Log-normal distribution

4516: 3912: 4511:{\displaystyle {\begin{aligned}f_{X}(x)&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl }\\&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl }\\&={\frac {\rm {d}}{{\rm {d}}x}}\operatorname {\Phi } \!\!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\&=\operatorname {\varphi } \!\left({\frac {\ln x-\mu }{\sigma }}\right){\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\&=\operatorname {\varphi } \!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right){\frac {1}{\ \sigma \ x\ }}\\&={\frac {1}{\ x\ \sigma {\sqrt {2\ \pi \ }}\ }}\exp \left(-{\frac {\ (\ln x-\mu )^{2}\ }{2\ \sigma ^{2}}}\right)~.\end{aligned}}} 7415: 14563: 21021: 19693: 14017: 6918: 7951: 7976: 6060: 23606: 20544: 19216: 2776: 7410:{\displaystyle {\begin{aligned}\operatorname {E} &=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}},\\\operatorname {E} &=e^{2\mu +2\sigma ^{2}},\\\operatorname {Var} &=\operatorname {E} -\operatorname {E} ^{2}=(\operatorname {E} )^{2}(e^{\sigma ^{2}}-1)=e^{2\mu +\sigma ^{2}}(e^{\sigma ^{2}}-1),\\\operatorname {SD} &={\sqrt {\operatorname {Var} }}=\operatorname {E} {\sqrt {e^{\sigma ^{2}}-1}}=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}{\sqrt {e^{\sigma ^{2}}-1}},\end{aligned}}} 21847: 20220: 7605: 14558:{\displaystyle {\begin{aligned}S_{+}&=\operatorname {E} \left=\sum _{i}\operatorname {E} =\sum _{i}e^{\mu _{i}+\sigma _{i}^{2}/2}\\\sigma _{Z}^{2}&=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}\operatorname {E} \operatorname {E} =1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}e^{\mu _{i}+\sigma _{i}^{2}/2}e^{\mu _{j}+\sigma _{j}^{2}/2}\\\mu _{Z}&=\ln \left(S_{+}\right)-\sigma _{Z}^{2}/2\end{aligned}}} 28118: 102: 41: 9602: 22988: 21016:{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}} 19688:{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}} 21527: 28128: 9690: 20539: 7946:{\displaystyle {\begin{aligned}\mu &=\ln \left({\frac {\operatorname {E} ^{2}}{\sqrt {\operatorname {E} }}}\right)=\ln \left({\frac {\operatorname {E} ^{2}}{\sqrt {\operatorname {Var} +\operatorname {E} ^{2}}}}\right),\\\sigma ^{2}&=\ln \left({\frac {\operatorname {E} }{\operatorname {E} ^{2}}}\right)=\ln \left(1+{\frac {\operatorname {Var} }{\operatorname {E} ^{2}}}\right).\end{aligned}}} 19846: 21626:, and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For example, the log-normal function with such 8912: 11039: 22976:. Hence, the approximation we have is in the second step (of the delta method), but the CI are actually for the expectation (not just the median). This is because we are starting from a base distribution that is normal, and then using another approximation after the log again to normal. This means that a big approximation part of the CI is from the delta method. 23601:{\displaystyle {\widehat {\left}}=\left{\frac {2}{\widehat {\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}}=\left{\frac {2}{{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}} 19207: 18474: 21047: 16367: 20215:{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\sim N\left((\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2}),{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)} 20235: 2248: 6002: 18947: 1957: 21706:, after Robert Gibrat (1904–1980) who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal. Consequently, 6715: 9597:{\displaystyle {\begin{aligned}E&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left}{\Phi \left}}\\E&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left}{1-\Phi \left}}\\E]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left-\Phi \left}{\Phi \left-\Phi \left}}\end{aligned}}} 15862: 12470: 18079: 8813: 14002: 21759:

For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of

22297:

to say that their logs is itself (approximately) normal. This trick allows us to pretend that their exp was log normal, and use that approximation to build the CI. Notice that in the RR case, the median and the mean in the base distribution (i.e., before taking the log), is actually identical (since

21713:

A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a

13356:

at the right tail. Its probability density function at the neighborhood of 0 has been characterized and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematically justified by Marlow) is obtained by

11051:

Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM and PopED. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different

22229:

analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main

17503: 10724: 21805:

In neuroscience, the distribution of firing rates across a population of neurons is often approximately log-normal. This has been first observed in the cortex and striatum and later in hippocampus and entorhinal cortex, and elsewhere in the brain. Also, intrinsic gain distributions and synaptic

15522: 21697:

The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting

17320: 25086: 18967: 13676: 510: 2080: 16183: 21717:

Specific examples are given in the following subsections. contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas. is a review article on log-normal distributions in neuroscience, with annotated bibliography.

18260: 18265: 21522:{\displaystyle E\left=E\left=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}} 1414: 5291: 4855: 10697: 10353: 1790: 10491: 17918: 20534:{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}} 9847: 2094: 9991: 3274: 10144: 8299: 15704: 5794: 18755: 1803: 13741: 24095: 15250: 4928: 2370: 1675: 1521: 24352:"Head-to-head, randomised, crossover study of oral versus subcutaneous methotrexate in patients with rheumatoid arthritis: drug-exposure limitations of oral methotrexate at doses >=15 mg may be overcome with subcutaneous administration" 22230:

use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.

776: 15321: 6572: 3377: 7583:

This estimate is sometimes referred to as the "geometric CV" (GCV), due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

12318: 22703: 17930: 11351: 11129: 8642: 672: 17805: 5727: 11034:{\displaystyle P(x;{\boldsymbol {\mu _{N}}},{\boldsymbol {\sigma _{N}}})={\frac {1}{x{\sqrt {2\pi \ln \left(1+\sigma _{N}^{2}/\mu _{N}^{2}\right)}}}}\exp \left(-{\frac {{\Big }^{2}}{2\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}}\right)} 3646: 11806: 11450: 17331: 17123: 3917: 17666: 205: 22547: 13360: 5121: 13220: 12293: 11933: 6901: 11658: 22124:

of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as

17166: 6268:

The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method.

4611: 13030: 5477:

within the integral. However, the log-normal distribution is not determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero. Indeed, the expected value

21999: 6923: 916: 19202:{\displaystyle {\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})}} 22190:, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." Also, the random obstruction of radio signals due to large buildings and hills, called 22078: 22402: 16386:

when analyzing log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.

14911: 14617: 13083: 11859: 11737: 11589: 7510: 8917: 8550: 7578: 5475: 1153: 5395: 16916: 385: 6415: 1963: 21596:

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution. This relationship is determined by the base of natural logarithm,

14955: 13147: 12200: 8108: 7610: 3794: 18469:{\displaystyle CI\left(E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}\right):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}} 11272: 4694: 1037: 18132: 14806: 14860: 8408: 14750: 17569: 12960: 1273: 1260: 22881: 15699: 6470: 6353: 6281:

Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed. (

5136: 4705: 19758: 14022: 13746: 13365: 269: 11526: 6525: 10514: 15316: 8634: 840: 18750:

Comparing the medians of the two can easily be done by taking the log from each and then constructing straightforward confidence intervals and transforming it back to the exponential scale.

10159: 1692: 21592: 18577: 6765: 18123: 16362:{\displaystyle \mu =\ln \left({\frac {\bar {x}}{\sqrt {1+{\widehat {\sigma }}^{2}/{\bar {x}}^{2}}}}\right),\qquad \sigma ^{2}=\ln \left(1+{{\widehat {\sigma }}^{2}}/{\bar {x}}^{2}\right).} 15095: 10364: 15138: 17810: 3900: 3471: 19211:

Plugin in the estimators to each of these parameters yields also a log normal distribution, which means that the Cox Method, discussed above, could similarly be used for this use-case:

18691: 18640: 18525: 12683: 2243:{\displaystyle \ {\frac {1}{2}}e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {\ 1+\operatorname {erf} \left({\frac {\sigma }{\ {\sqrt {2\ }}\ }}+\operatorname {erf} ^{-1}(2p-1)\right)\ }{p}}} 21687: 16034: 372: 15972: 9711: 2452:

has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and

22974: 12117: 12071: 5997:{\displaystyle \varphi (t)\approx {\frac {\exp \left(-{\frac {W^{2}(-it\sigma ^{2}e^{\mu })+2W(-it\sigma ^{2}e^{\mu })}{2\sigma ^{2}}}\right)}{\sqrt {1+W(-it\sigma ^{2}e^{\mu })}}}} 12888: 9858: 3425: 3169: 22019: 18942:{\displaystyle CI(e^{\mu _{1}-\mu _{2}}):e^{\left({\hat {\mu }}_{1}-{\hat {\mu }}_{2}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n}}+{\frac {S_{2}^{2}}{n}}}}\right)}} 16138: 11188: 10006: 9676: 8175: 3164: 13736: 13314: 7967:. That is, there exist other distributions with the same set of moments. In fact, there is a whole family of distributions with the same moments as the log-normal distribution. 7453: 6549: 2818: 2681: 1952:{\displaystyle \ \mu =\log \left({\frac {\operatorname {\mathbb {E} } \ }{\ {\sqrt {{\frac {\ \operatorname {Var} ~~}{\ \operatorname {\mathbb {E} } ^{2}\ }}+1\ }}\ }}\right)\ ,} 16106: 8338: 6119: 5589: 5518: 968: 24629:

Retout, S; Duffull, S; Mentré, F (2001). "Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs".

17598: 16968: 14688: 5028: 5006: 4984: 21819:

In the size of avalanches of fractures in the cytoskeleton of living cells, showing log-normal distributions, with significantly higher size in cancer cells than healthy ones.

3728: 2951: 2906: 306: 8044: 6231: 3545: 2857: 22739: 21624: 6160: 2992: 12556: 9636: 3849: 3504: 3093: 2636: 12710: 5766: 5607:

at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor

4881: 2255: 16063: 12752: 3061: 1547: 136: 19822: 19790: 15579: 4640: 3823: 3025: 1427: 22107: 15270: 11959: 6710:{\displaystyle \operatorname {E} =e^{\mu +{\frac {1}{2}}\sigma ^{2}}=e^{\mu }\cdot {\sqrt {e^{\sigma ^{2}}}}=\operatorname {GM} \cdot {\sqrt {\operatorname {GVar} }}.} 6049: 3123: 70: 21948: 21710:

for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.

21644: 21557: 16178: 15928: 15908: 15619: 13240: 12137: 11684: 8147: 6192: 2724: 2610: 685: 90: 21867:, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes. 21714:

solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.

15857:{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad {\widehat {\sigma }}^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n}}.} 12610: 12465:{\displaystyle Y=\textstyle \prod _{j=1}^{n}X_{j}\sim \operatorname {Lognormal} {\Big (}\textstyle \sum _{j=1}^{n}\mu _{j},\ \sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.} 6257: 18745: 18718: 18074:{\displaystyle {\widehat {\mu }}+{\frac {S^{2}}{2}}{\dot {\sim }}N\left(\mu +{\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{2}}{n}}+{\frac {\sigma ^{4}}{2(n-1)}}\right)} 17694: 15552: 13703: 12530: 12025: 11998: 8836: 3672: 3579: 3279: 2770: 2566: 23805: 23671: 17154: 16803: 16158: 15888: 15599: 13260: 8808:{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi \!\left({\frac {\mu +\sigma ^{2}-\ln k}{\sigma }}\right)} 6531:

to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of

3692: 2704: 2590: 14643: 28167: 22552: 13997:{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left,\\\mu _{Z}&=\ln \!\left+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}} 26587:

Alamsar, Mohammed; Parisis, George; Clegg, Richard; Zakhleniuk, Nickolay (2019). "On the Distribution of Traffic Volumes in the Internet and its Implications".

18962:

However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop. The ratio of their means is:

16768: 16610: 13354: 13334: 12842: 12822: 12802: 12503: 12313: 12220: 11280: 11058: 8904: 8884: 8456: 8436: 8167: 6081: 6025: 5786: 5538: 523: 25939:

Polizzi, Stefano; Laperrousaz, Bastien; Perez-Reche, Francisco J; Nicolini, Franck E; Satta, Véronique Maguer; Arneodo, Alain; Argoul, Françoise (2018-05-29).

21559:

is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that

17049: 16864: 16491: 16441: 17715: 14958: 5617: 17498:{\displaystyle CI(E(X)):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}} 3584: 23838: 11742: 26776: 17054: 16770:

contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.

11356: 25433:

Pawel, Sobkowicz; et al. (2013). "Lognormal distributions of user post lengths in Internet discussions - a consequence of the Weber-Fechner law?".

25252:

Bloetscher, Frederick (2019). "Using predictive Bayesian Monte Carlo- Markov Chain methods to provide a probabilistic solution for the Drake equation".

17603: 15517:{\displaystyle \ell (\mu ,\sigma \mid x_{1},x_{2},\ldots ,x_{n})=-\sum _{i}\ln x_{i}+\ell _{N}(\mu ,\sigma \mid \ln x_{1},\ln x_{2},\dots ,\ln x_{n}).} 148: 22407: 5036: 22705:. So while we expect the CI to be for the median, in this case, it's actually also for the mean in the original distribution. i.e., if the original 13155: 12228: 11864: 18959:. The way it is done there is that we have two approximately Normal distributions (e.g., p1 and p2, for RR), and we wish to calculate their ratio. 17315:{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad S^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n-1}}} 6906:

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable

6815: 26264:"A Preferential Attachment Paradox: How Preferential Attachment Combines with Growth to Produce Networks with Log-normal In-degree Distributions" 25142:

Wu, Zi-Niu (2003). "Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall".

25068:

Land, C. E. (1971), “Confidence intervals for linear functions ofthe normal mean and variance,” Annals of Mathematical Statistics, 42, 1187–1205.

11594: 22237:

when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.

4536: 23738: 12969: 24329: 21957: 853: 16076:

can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation

26361: 5544: 1539: 24277:"Onset-Duration Matching of Acoustic Stimuli Revisited: Conventional Arithmetic vs. Proposed Geometric Measures of Accuracy and Precision" 26905: 26424: 26377: 22028: 6282: 6270: 25465: 22301: 17924: 14865: 14571: 13671:{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left,\\\mu _{Z}&=\ln \!\left-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}} 13037: 11813: 11691: 11543: 7458: 28131: 27388: 8464: 23899: 22284:

The Cox Method was quoted as “personal communication” in Land, 1971, and was also given in CitationZhou and Gao (1997) and Olsson 2005

7518: 5403: 1050: 505:{\displaystyle \ {\frac {1}{\ x\sigma {\sqrt {2\pi \ }}\ }}\ \exp \left(-{\frac {\left(\ln x-\mu \ \right)^{2}}{2\sigma ^{2}}}\right)} 28162: 27296: 22257: 5314: 2075:{\displaystyle \ \sigma ={\sqrt {\log \left({\frac {\ \operatorname {Var} ~~}{\ \operatorname {\mathbb {E} } ^{2}\ }}+1\ \right)\ }}} 25628:

Makuch, Robert W.; D.H. Freeman; M.F. Johnson (1979). "Justification for the lognormal distribution as a model for blood pressure".

21562: 16869: 28083: 24811: 14974: 14964:

A substitute for the log-normal whose integral can be expressed in terms of more elementary functions can be obtained based on the

2514: 23629:"Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" 22162:(population) satisfy Gibrat's Law. The growth process of city sizes is proportionate and invariant with respect to size. From the 21878:, illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% 18255:{\displaystyle {\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}} 6362: 5788:

in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by

27949: 27161: 26920: 26769: 21813: 14916: 13088: 12142: 8839: 8052: 3735: 4645: 981: 27844: 27608: 14755: 11193: 1409:{\displaystyle \ 1\ \exp \left(4\ \sigma ^{2}\right)+2\ \exp \left(3\ \sigma ^{2}\right)+3\ \exp \left(2\sigma ^{2}\right)-6\ } 22130: 18587:

Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in an

14821: 8349: 5286:{\displaystyle \operatorname {Var} _{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}(e^{\Sigma _{ij}}-1).} 27282: 26638: 26408: 26065: 25499: 24898: 24416: 23961: 23935: 23781: 14693: 4850:{\displaystyle {\frac {1}{2}}\left={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)} 21780:

Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).

17512: 12913: 1166: 27603: 27547: 27445: 27207: 26845: 25041:

Olsson, Ulf. "Confidence intervals for the mean of a log-normal distribution." Journal of Statistics Education 13.1 (2005).

22744: 15624: 6423: 6300: 23872: 21646:

fits well with the size of secondarily produced droplets during droplet impact and the spreading of an epidemic disease.

17325: 10692:{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma _{g}}})={\frac {1}{x\ln(\sigma _{g}){\sqrt {2\pi }}}}\exp \left} 27889: 27623: 27476: 27151: 26895: 24871:

Botev, Z. I.; L'Ecuyer, P. (2017). "Accurate computation of the right tail of the sum of dependent log-normal variates".

19829: 19705: 17701: 12894: 218: 27353: 23772:, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: 23628: 21740:

Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.

10348:{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {cv}})={\frac {1}{x{\sqrt {\ln(cv^{2}+1)}}{\sqrt {2\pi }}}}\exp \left} 6479: 1785:{\displaystyle \ {\begin{pmatrix}{\frac {1}{\ \sigma ^{2}\ }}&0\\0&{\frac {2}{\ \sigma ^{2}\ }}\end{pmatrix}}\ } 28121: 27793: 27769: 27348: 26762: 25056:

user10525, How do I calculate a confidence interval for the mean of a log-normal data set?, URL (version: 2022-12-18):

24551:

Lunn, D. (2012). The BUGS book: a practical introduction to Bayesian analysis. Texts in statistical science. CRC Press.

11455: 26493:

Gros, C; Kaczor, G.; Markovic, D (2012). "Neuropsychological constraints to human data production on a global scale".

15275: 14014:

The sum of correlated log-normally distributed random variables can also be approximated by a log-normal distribution

10486:{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}{\frac {1}{x}}\exp \left} 8562: 2569: 782: 28152: 27990: 27867: 27828: 27800: 27774: 27692: 27618: 27041: 26789: 26188: 24458: 21923:

of 97%–99% of the population is distributed log-normally. (The distribution of higher-income individuals follows a

25077:

Zhou, X-H., and Gao, S. (1997), “Confidence intervals for the log-normal mean,” Statistics in Medicine, 16, 783–790.

18530: 17913:{\displaystyle {\frac {S^{2}}{2}}{\dot {\sim }}N\left({\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{4}}{2(n-1)}}\right)} 6723: 6031:. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of 27978: 27944: 27810: 27805: 27650: 27458: 27156: 26910: 24914:

Asmussen, A.; Goffard, P.-O.; Laub, P. J. (2016). "Orthonormal polynomial expansions and lognormal sum densities".

22159: 18088: 14969: 6473: 4931: 4527: 516: 15621:. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations 15128: 28172: 27728: 27641: 27613: 27522: 27471: 27343: 27126: 27091: 24051: 3854: 3430: 26688: 26671: 25345: 25328: 25179:"Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model" 24698: 24579: 24562: 24257: 24240: 18645: 18594: 18490: 12715:

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of

12615: 9842:{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left} 2464:

and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

27742: 27659: 27496: 27420: 27243: 27121: 27096: 26960: 26955: 26950: 24129:

S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution",

21767:

The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.

21042:, yet biased, point-estimation (we use the fact that the estimator of the ratio is a log normal distribution): 16073: 15980: 1796: 324: 21698:

accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "

21652: 20225: 18082: 15941: 11531:

All remaining re-parameterisation formulas can be found in the specification document on the project website.

9986:{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{x{\sqrt {v}}{\sqrt {2\pi }}}}\exp \left} 6776: 5296:

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the

3269:{\displaystyle \ \mu =\ln \left({\frac {\mu _{X}^{2}}{\ {\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}\ }}\ }}\right)\ } 28058: 27924: 27632: 27481: 27413: 27398: 27291: 27265: 27197: 27036: 26930: 26925: 26867: 26852: 26056:. Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 23636: 21900: 21031: 10139:{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left} 8294:{\displaystyle q_{X}(\alpha )=e^{\mu +\sigma q_{\Phi }(\alpha )}=\mu ^{*}(\sigma ^{*})^{q_{\Phi }(\alpha )},} 3851:

be respectively the cumulative probability distribution function and the probability density function of the

24663:

The PopED Development Team (2016). PopED Manual, Release version 2.13. Technical report, Uppsala University.

22886: 21770:

The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.

12076: 12030: 8003:

is the point of global maximum of the probability density function. In particular, by solving the equation

28157: 27894: 27884: 27575: 27501: 27202: 27061: 22252: 14812: 12850: 12778: 10502: 8843: 6356: 4867: 3903: 3384: 378: 27954: 24725: 22004: 16111: 9641: 3128: 27939: 27934: 27879: 27815: 27759: 27580: 27567: 27358: 27303: 27255: 27046: 26975: 26840: 26217:

Thelwall, Mike; Wilson, Paul (2014). "Regression for citation data: An evaluation of different methods".

24133: 16923: 16377: 13708: 11134: 8149:

has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of

7426: 6534: 2782: 2644: 25536:

S. K. Chan, Jennifer; Yu, Philip L. H. (2006). "Modelling SARS data using threshold geometric process".

25023: 24844:

Marlow, NA. (Nov 1967). "A normal limit theorem for power sums of independent normal random variables".

22145:. The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the 21809:

Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.

16079: 13265: 8307: 6086: 5549: 5481: 4986:

has a multivariate log-normal distribution. The exponential is applied elementwise to the random vector

929: 28073: 27849: 27668: 27450: 27403: 27272: 27248: 27228: 27071: 26945: 26825: 26633:, Statistics: Textbooks and Monographs, vol. 88, New York: Marcel Dekker, Inc., pp. xvi+387, 21829: 19837: 17709: 17574: 17157: 16933: 16615: 16496: 14652: 5011: 4989: 4937: 1527: 26097:"Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States" 16973: 6233:(blue regions), using the numerical method of ray-tracing. b & c. The pdf and cdf of the function 3697: 2911: 2866: 279: 28078: 27862: 27823: 27697: 27534: 27378: 27323: 27221: 27185: 27056: 27021: 25042: 24594:

Nyberg, J.; et al. (2012). "PopED - An extended, parallelized, population optimal design tool".

22247: 22156:, the number of citations to journal articles and patents follows a discrete log-normal distribution. 21799: 17923:

Since the sample mean and variance are independent, and the sum of normally distributed variables is

15245:{\displaystyle L(\mu ,\sigma )=\prod _{i=1}^{n}{\frac {1}{x_{i}}}\varphi _{\mu ,\sigma }(\ln x_{i}),} 15098: 13336:

has no closed-form expression, but can be reasonably approximated by another log-normal distribution

8855: 6197: 3517: 2823: 17: 26722:(1994). "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion". 24448: 23825: 23730: 22708: 21600: 6124: 4923:{\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})} 2365:{\displaystyle \ =e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {1}{1-p}}(1-\Phi (\Phi ^{-1}(p)-\sigma ))} 101: 40: 27764: 27552: 27318: 27277: 27192: 27146: 27086: 27051: 26940: 26835: 26785: 26616:

ASTM D4577, Standard Test Method for Compression Resistance of a container Under Constant Load>\

25998:"The lognormal distribution of the elements (A fundamental law of geochemistry and its subsidiary)" 25665:"Common noncompartmental pharmacokinetic variables: are they normally or log-normally distributed?" 25451:

Silence is also evidence: interpreting dwell time for recommendation from psychological perspective

24935:

Sangal, B.; Biswas, A. (1970). "The 3-Parameter Lognormal Distribution Applications in Hydrology".

24326: 22183:

analysis, the log-normal distribution is often used to model times to repair a maintainable system.

22180: 21039: 19825: 17697: 11970: 10151: 9683: 8114: 7421: 6552: 2956: 2499: 2394: 1670:{\displaystyle \ \sum _{n=0}^{\infty }{\frac {\ (i\ t)^{n}\ }{n!}}e^{\ n\mu +n^{2}\sigma ^{2}/2}\ } 25580: 12535: 9615: 3828: 3476: 3066: 2615: 1516:{\displaystyle \ \log _{2}\left(\ {\sqrt {2\pi \ }}\ \sigma \ e^{\mu +{\tfrac {1}{2}}}\ \right)\ } 28063: 28005: 27676: 27463: 27373: 27313: 27131: 27081: 27076: 26877: 26857: 25664: 22117: 21833: 12688: 8556: 5742: 5297: 27233: 26057: 26051: 22219:

File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.

16039: 12754:

to all have finite variance and satisfy the other conditions of any of the many variants of the

12718: 8006: 3030: 2863:

This relationship is true regardless of the base of the logarithmic or exponential function: If

2480: 771:{\displaystyle \ \exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\right)\ } 109: 27929: 27917: 27906: 27788: 27684: 27491: 26935: 26915: 26820: 25449: 24685:

Damgaard, Christian; Weiner, Jacob (2000). "Describing inequality in plant size or fecundity".

23820: 22293:

The issue is that we don't know how to do it directly, so we take their logs, and then use the

22187: 19795: 19763: 15557: 4619: 3802: 2997: 26432: 25940: 24130: 22083: 21727:

The length of comments posted in Internet discussion forums follows a log-normal distribution.

15255: 11938: 6034: 3372:{\displaystyle \ \sigma ^{2}=\ln \left(1+{\frac {\ \sigma _{X}^{2}\ }{\mu _{X}^{2}}}\right)~.} 3101: 49: 28053: 28010: 27854: 27529: 27383: 27363: 27260: 26830: 25727:"Preconfigured, skewed distribution of firing rates in the hippocampus and entorhinal cortex" 22163: 22146: 21933: 21883: 21629: 21542: 19833: 17705: 16163: 15913: 15893: 15604: 14965: 13225: 12755: 12122: 11663: 8120: 6165: 2709: 2595: 2506: 317: 211: 75: 12561: 6236: 28103: 28098: 28093: 28088: 28025: 27995: 27874: 27517: 27408: 27011: 26970: 26965: 26862: 26735: 26648: 26512: 26459: 26009: 25952: 25261: 25200: 25151: 25114: 24962: 24024: 23988:

Holgate, P. (1989). "The lognormal characteristic function, vol. 18, pp. 4539–4548, 1989".

23890: 23791: 23773: 23703: 22698:{\displaystyle CI(p_{1}):e^{log({\hat {p}}_{1})\pm (1-{\hat {p}}_{1})/({\hat {p}}_{1}*n))}} 21851: 21795: 21667: 18723: 18696: 17672: 15537: 13681: 12508: 12003: 11976: 8821: 8343:

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,

6800: 6768: 6559: 3651: 3558: 2749: 2545: 2495: 2431: 27308: 26656: 25103:"Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy" 23768:

Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), "14: Lognormal Distributions",

21730:

Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.

17139: 16788: 16143: 15873: 15584: 13245: 11964: 11346:{\displaystyle \operatorname {LN7} (\mu _{N},\sigma _{N})\to \operatorname {LN2} (\mu ,v)} 11124:{\displaystyle \operatorname {LN2} (\mu ,v)\to \operatorname {LN7} (\mu _{N},\sigma _{N})} 8906:—is its partial expectation divided by the cumulative probability of being in that range: 3677: 2689: 2575: 667:{\displaystyle \ {\frac {\ 1\ }{2}}\left=\Phi \left({\frac {\ln(x)-\mu }{\sigma }}\right)} 8: 28037: 27562: 27542: 27512: 27486: 27440: 27368: 27180: 27116: 22213: 21924: 21788: 21774: 20541:

And since confidence intervals are preserved for monotonic transformations, we get that:

18262:

And since confidence intervals are preserved for monotonic transformations, we get that:

17133: 16782: 16396: 15132: 14622: 12963: 12898: 6772: 2406: 1420: 26516: 26463: 26013: 25956: 25916: 25891: 25265: 25204: 25155: 25118: 24028: 22169:

The number of sexual partners appears to be best described by a log-normal distribution.

17800:{\displaystyle S^{2}{\dot {\sim }}N\left(\sigma ^{2},{\frac {2\sigma ^{4}}{n-1}}\right)} 28068: 27557: 27338: 27333: 27238: 27175: 27170: 27026: 27016: 26900: 26588: 26528: 26502: 26342: 26298: 26275: 26263: 26244: 26226: 26159: 26114: 25867: 25840: 25816: 25783: 25759: 25726: 25561: 25405: 25372: 25304: 25277: 25229: 25190: 25178: 24984: 24915: 24876: 24857: 24517: 24492: 24436: 24427: 24376: 24351: 24303: 24276: 24221: 24186: 24163: 24151: 23663: 23645: 22138: 21842:

Diameters of crystals in ice cream, oil drops in mayonnaise, pores in cocoa press cake.

21761: 16443:

contains approximately two thirds (68%) of the probability (or of a large sample), and

16066: 15116: 14008: 13339: 13319: 12827: 12807: 12787: 12488: 12298: 12205: 9703: 8889: 8869: 8441: 8421: 8152: 6066: 6028: 6010: 5771: 5722:{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}} 5523: 2735: 2378: 2086: 1685: 272: 25997: 25163: 24642: 16808: 16446: 16402: 27966: 27393: 27136: 27066: 27031: 26980: 26731: 26634: 26569: 26475: 26404: 26381: 26303: 26184: 26163: 26109:

Wataru, Souma (2002-02-22). "Physics of Personal Income". In Takayasu, Hideki (ed.).

26061: 26025: 26021: 25978: 25921: 25872: 25821: 25803: 25764: 25746: 25692: 25684: 25645: 25641: 25610: 25602: 25553: 25505: 25495: 25410: 25392: 25350: 25281: 25234: 25216: 24992: 24894: 24756: 24646: 24611: 24522: 24454: 24412: 24381: 24308: 24191: 23967: 23957: 23931: 23777: 23667: 22126: 21896: 16383: 8000: 7988: 6558:

Note that the geometric mean is smaller than the arithmetic mean. This is due to the

5604: 5127: 3641:{\displaystyle \ X\sim \operatorname {Lognormal} \left(\ \mu ,\sigma ^{2}\ \right)\ } 974: 678: 26532: 25892:"Ubiquitous lognormal distribution of neuron densities in mammalian cerebral cortex" 25799: 25784:"Population-wide distributions of neural activity during perceptual decision-making" 25597: 25565: 25522:

Sartwell, Philip E. "The distribution of incubation periods of infectious disease."

25273: 24508: 24367: 24015:

Barakat, R. (1976). "Sums of independent lognormally distributed random variables".

22197:

Particle size distributions produced by comminution with random impacts, such as in

21930:

If an income distribution follows a log-normal distribution with standard deviation

11801:{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Lognormal} (-\mu ,\ \sigma ^{2}).} 11445:{\textstyle \mu =\ln \left(\mu _{N}/{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}\right)} 27141: 26815: 26754: 26749: 26706: 26683: 26652: 26559: 26520: 26467: 26373: 26334: 26293: 26285: 26248: 26236: 26151: 26142:

Black, F.; Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities".

26124: 26096: 26017: 25968: 25960: 25911: 25903: 25862: 25857: 25852: 25811: 25795: 25754: 25738: 25676: 25637: 25592: 25545: 25400: 25384: 25340: 25269: 25224: 25208: 25159: 25122: 25019: 24976: 24944: 24886: 24853: 24826: 24790: 24752: 24748: 24739:

Thorin, Olof (1977). "On the infinite divisibility of the lognormal distribution".

24721: 24694: 24638: 24603: 24574: 24512: 24504: 24422: 24404: 24371: 24363: 24298: 24288: 24252: 24181: 24173: 24110: 24074: 24066: 24032: 23997: 23921: 23830: 23655: 22226: 21951: 21850:

Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see

21784: 21707: 20229: 18126: 17118:{\displaystyle \operatorname {sem} ^{*}=({\widehat {\sigma }}^{*})^{1/{\sqrt {n}}}} 6563: 5733: 4642:

is the cumulative distribution function of the standard normal distribution (i.e.,

2488: 23834: 26644: 26524: 26447: 26178: 26092: 25742: 24333: 23787: 21879: 21703: 21035: 18591:). We have samples from two independent log-normal distributions with parameters 17661:{\displaystyle {\widehat {\mu }}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right)} 12762: 12480: 5603:

that has a negative imaginary part, and hence the characteristic function is not

2510: 2502: 2484: 2479:. The log-normal distribution has also been associated with other names, such as 2398: 1266: 309: 200:{\displaystyle \ \operatorname {Lognormal} \left(\ \mu ,\,\sigma ^{2}\ \right)\ } 26607:

ASTM D3654, Standard Test Method for Shear Adhesion on Pressure-Sensitive Tapesw

23926: 23863: 6420:

By analogy with the arithmetic statistics, one can define a geometric variance,

27214: 26289: 25964: 24779:"Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables" 24607: 23659: 22542:{\displaystyle log({\hat {p}}_{1}){\dot {\sim }}N(log(p_{1}),(1-p1)/(p_{1}*n))} 22153: 22022: 9678:, here are multiple ways how the log-normal distribution can be parameterized. 7599:

can be obtained, if the arithmetic mean and the arithmetic variance are known:

6294: 5116:{\displaystyle \operatorname {E} _{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},} 2727: 2476: 846: 26710: 26240: 26128: 25680: 25212: 24830: 24212:

Kirkwood, Thomas BL (Dec 1979). "Geometric means and measures of dispersion".

24115: 24001: 14011:

to estimate the cumulative distribution function, the pdf and the right tail.

13215:{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})} 12288:{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})} 11928:{\displaystyle X^{a}\sim \operatorname {Lognormal} (a\mu ,\ a^{2}\sigma ^{2})} 7975: 6059: 28146: 27837: 27585: 26872: 26573: 26479: 26448:"Packing Densities of Mixtures of Spheres with Log-normal Size Distributions" 26338: 26029: 25982: 25907: 25807: 25750: 25688: 25606: 25509: 25396: 25354: 25220: 24890: 24760: 24712:

Rossman, Lewis A (July 1990). "Design stream flows based on harmonic means".

24293: 23971: 22234: 18952: 5608: 26378:

10.1002/(SICI)1097-0258(19960130)15:2<221::AID-SIM148>3.0.CO;2-Q

24948: 21531: 6896:{\displaystyle \operatorname {E} =e^{n\mu +{\frac {1}{2}}n^{2}\sigma ^{2}}.} 26471: 26307: 26044: 25925: 25876: 25825: 25768: 25614: 25557: 25414: 25373:"The log-dynamic brain: how skewed distributions affect network operations" 25238: 25045: 24996: 24965:(1949). "Systems of Frequency Curves Generated by Methods of Translation". 24650: 24615: 24542:

Forbes et al. Probability Distributions (2011), John Wiley & Sons, Inc.

24526: 24385: 24312: 24195: 24036: 22549:

Hence, building a CI based on the log and than back-transform will give us

22294: 16930:

degrees of freedom. Back-transformation leads to a confidence interval for

12774: 12773:

A set of data that arises from the log-normal distribution has a symmetric

11653:{\displaystyle aX\sim \operatorname {Lognormal} (\mu +\ln a,\ \sigma ^{2})} 26564: 26547: 26385: 25890:

Morales-Gregorio, Aitor; van Meegen, Alexander; van Albada, Sacha (2023).

25696: 25057: 24795: 24778: 24408: 24070: 23951: 22109:

is the cumulative distribution function of a standard normal distribution.

18951:

These CI are what's often used in epidemiology for calculation the CI for

18579:, which seemed to provide better coverage results for small sample sizes. 4606:{\displaystyle F_{X}(x)=\Phi \left({\frac {(\ln x)-\mu }{\sigma }}\right)} 2494:

A log-normal process is the statistical realization of the multiplicative

26719: 26119: 25649: 22134: 18588: 13025:{\displaystyle \exp(X)\sim \operatorname {Lognormal} (\mu ,\sigma ^{2}).} 8854:, it is used in solving the partial differential equation leading to the 8636:. For a log-normal random variable, the partial expectation is given by: 5303: 2775: 2453: 26080: 25782:

Wohrer, Adrien; Humphries, Mark D.; Machens, Christian K. (2013-04-01).

24812:"Asymptotics of Sums of Lognormal Random Variables with Gaussian Copula" 24403:. Vol. 46 (5th ed.). Oxford, UK: Wiley-Blackwell. p. 89. 24177: 24079: 13222:

be independent log-normally distributed variables with possibly varying

26346: 26322: 26111:

Empirical Science of Financial Fluctuations: The Advent of Econophysics

25308: 24988: 24967: 24225: 22222:

Sizes of text-based emails (1990s) and multimedia-based emails (2000s).

22209: 21994:{\displaystyle G=\operatorname {erf} \left({\frac {\sigma }{2}}\right)} 18956: 11965:

Multiplication and division of independent, log-normal random variables

911:{\displaystyle \ \exp \left(\ \mu +{\frac {\sigma ^{2}}{2}}\ \right)\ } 25973: 25127: 25102: 22137:

would be a more appropriate model, in particular for the analysis for

19702:

To construct a confidence interval for this ratio, we first note that

10359:

LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale

10001:, m, on the natural scale and standard deviation, σ, on the log-scale 6276: 25938: 25549: 24920: 24327:"FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS" 23806:"Maximum entropy autoregressive conditional heteroskedasticity model" 22205: 22198: 21916: 21864: 15910:

is biased. As for the normal distribution, an unbiased estimator for

9853:

LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale

8851: 8847: 7956:

A probability distribution is not uniquely determined by the moments

2739: 2461: 2402: 25889: 25841:"Logarithmic distributions prove that intrinsic learning is Hebbian" 25663:

Lacey, L. F.; Keene, O. N.; Pritchard, J. F.; Bye, A. (1997-01-01).

25388: 24980: 24493:"ProbOnto: ontology and knowledge base of probability distributions" 21689:

is used to provide a probabilistic solution for the Drake equation.

18582: 26593: 26280: 26231: 26155: 24881: 24168: 23650: 22142: 22073:{\displaystyle G=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1} 13357:

matching the mean and variance of another log-normal distribution:

12027:

are multiplied , the product is again log-normal, with parameters

9679: 7992: 2457: 1159: 1043: 26507: 25195: 24270: 24268: 23916:

Heyde, CC. (2010), "On a Property of the Lognormal Distribution",

23898:. Casualty Actuarial Society E-Forum, Spring 2015. Arlington, VA. 22397:{\displaystyle {\hat {p}}_{1}{\dot {\sim }}N(p_{1},p_{1}(1-p1)/n)} 21846: 14906:{\displaystyle Y\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})} 14811:

The log-normal distribution is a special case of the semi-bounded

14612:{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})} 13078:{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})} 12844:

means of this distribution are related; such relation is given by

11854:{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})} 11732:{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})} 11584:{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})} 7505:{\displaystyle {\tfrac {\operatorname {SD} }{\operatorname {E} }}} 2505:, each of which is positive. This is justified by considering the 24131:

Methodology and Computing in Applied Probability 18 (2), 441-458.

22113: 21954:, commonly use to evaluate income inequality, can be computed as 21875: 8545:{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx.} 26362:"The Shape of the Distribution of the Number of Sexual Partners" 24096:"On Lognormal Random Variables: I – The Characteristic Function" 12505:

independent, identically distributed, positive random variables

7573:{\displaystyle \operatorname {CV} ={\sqrt {e^{\sigma ^{2}}-1}}.} 5470:{\displaystyle z={\tfrac {\ln(x)-(\mu +n\sigma ^{2})}{\sigma }}} 1148:{\displaystyle \ \left\ \exp \left(2\ \mu +\sigma ^{2}\right)\ } 25581:"PBSIM: PacBio reads simulator—toward accurate genome assembly" 24265: 22262: 22208:

distribution of publicly available audio and video data files (

22191: 21920: 9998: 7984: 5390:{\displaystyle \operatorname {E} =e^{n\mu +n^{2}\sigma ^{2}/2}} 3509: 922: 25627: 25329:"Log-normal Distributions across the Sciences: Keys and Clues" 24783:

International Journal of Mathematics and Mathematical Sciences

24563:"Log-normal distributions across the sciences: Keys and clues" 21750:

Measures of size of living tissue (length, skin area, weight).

16911:{\displaystyle \mathop {se} ={\widehat {\sigma }}/{\sqrt {n}}} 9693:

Overview of parameterizations of the log-normal distributions.

9689: 3381:

Alternatively, the "multiplicative" or "geometric" parameters

26697:

Holgate, P. (1989). "The lognormal characteristic function".

26672:"Lognormal distributions across the sciences: keys and clues" 26586: 25579:

Ono, Yukiteru; Asai, Kiyoshi; Hamada, Michiaki (2013-01-01).

25177:

Wang, WenBin; Wu, ZiNiu; Wang, ChunFeng; Hu, RuiFeng (2013).

24241:"Lognormal distributions across the sciences: keys and clues" 21734: 17132:

The literature discusses several options for calculating the

25941:"A minimal rupture cascade model for living cell plasticity" 25448:

Yin, Peifeng; Luo, Ping; Lee, Wang-Chien; Wang, Min (2013).

8866:

The conditional expectation of a log-normal random variable

6410:{\displaystyle \operatorname {GSD} =e^{\sigma }=\sigma ^{*}} 6259:

of the log-normal variable can also be computed in this way.

26689:

10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2

25346:

10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2

24699:

10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2

24580:

10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2

24258:

10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2

23956:(Anniversary ed.). Hoboken, N.J.: Wiley. p. 415. 23627:

Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019).

18487:

Olsson 2005, proposed a "modified Cox method" by replacing

14950:{\displaystyle X\sim \operatorname {Suzuki} (\mu ,\sigma )} 13142:{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})} 12195:{\displaystyle \sigma ^{2}=\sigma _{1}^{2}+\sigma _{2}^{2}} 9699: 8103:{\displaystyle \operatorname {Mode} =e^{\mu -\sigma ^{2}}.} 7980: 3789:{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})} 2731: 25327:

Limpert, Eckhard; Stahel, Werner A.; Abbt, Markus (2001).

24470: 12558:, approximately a log-normal distribution with parameters 11267:{\textstyle \sigma _{N}=\exp(\mu +v/2){\sqrt {\exp(v)-1}}} 6194:, then integrating its density over the domain defined by 4689:{\displaystyle \ \operatorname {\mathcal {N}} (\ 0,\ 1)\ } 3547:

is useful for determining "scatter" intervals, see below.

1032:{\displaystyle \ \exp \left(\ \mu -\sigma ^{2}\ \right)\ } 24873:

2017 Winter Simulation Conference (WSC), 3rd–6th Dec 2017

24152:"A method to integrate and classify normal distributions" 21777:

sequencing read length follows a log-normal distribution.

21756:

Diameters of banana leaf spots, powdery mildew on barley.

21699: 17156:(the mean of the log-normal distribution). These include 14801:{\displaystyle \operatorname {Var} =\operatorname {Var} } 24103:

Journal of the Australian Mathematical Society, Series B

22166:

therefore, the log of city size is normally distributed.

22141:. Indeed, stock price distributions typically exhibit a 21532:

Extremal principle of entropy to fix the free parameter

14855:{\displaystyle X\mid Y\sim \operatorname {Rayleigh} (Y)} 8403:{\displaystyle \operatorname {Med} =e^{\mu }=\mu ^{*}~.} 2779:

Relation between normal and log-normal distribution. If

24809: 24732: 23767: 22298:

they are originally normal, and not log normal). E.g.,

16493:

contain 95%. Therefore, for a log-normal distribution,

14745:{\displaystyle \operatorname {E} =\operatorname {E} +c} 12474: 26750:

The normal distribution is the log-normal distribution

25781: 25662: 24673: 21655: 17564:{\displaystyle E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}} 13268: 12955:{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})} 12633: 12376: 12328: 12315:

independent, log-normally distributed variables, then

11747: 11458: 11359: 11196: 11137: 8846:. The partial expectation formula has applications in 8732: 7463: 7350: 6960: 6288: 5739:

A closed-form formula for the characteristic function

5520:

is not defined for any positive value of the argument

5414: 5304:

Characteristic function and moment generating function

2686:

is called the log-normal distribution with parameters

1704: 1489: 1255:{\displaystyle \ \left{\sqrt {\exp(\sigma ^{2})-1\;}}} 25010:

Swamee, P. K. (2002). "Near Lognormal Distribution".

24490: 24049: 23626: 22991: 22889: 22876:{\displaystyle E=e^{log(p_{1})+1/2*(1-p1)/(p_{1}*n)}} 22747: 22711: 22555: 22410: 22304: 22086: 22031: 22007: 21960: 21936: 21806:

weight distributions appear to be log-normal as well.

21632: 21603: 21565: 21545: 21050: 20547: 20238: 19849: 19798: 19766: 19708: 19219: 18970: 18758: 18726: 18699: 18648: 18597: 18533: 18493: 18268: 18135: 18091: 17933: 17813: 17718: 17675: 17606: 17577: 17515: 17334: 17169: 17142: 17057: 16976: 16936: 16872: 16811: 16791: 16618: 16499: 16449: 16405: 16186: 16166: 16146: 16114: 16082: 16042: 15983: 15944: 15916: 15896: 15876: 15707: 15694:{\displaystyle \ln x_{1},\ln x_{2},\dots ,\ln x_{n})} 15627: 15607: 15587: 15560: 15540: 15324: 15278: 15258: 15141: 15119:

estimators of the log-normal distribution parameters

14977: 14919: 14868: 14824: 14758: 14696: 14655: 14625: 14574: 14020: 13744: 13711: 13684: 13363: 13342: 13322: 13248: 13228: 13158: 13091: 13040: 12972: 12916: 12853: 12830: 12810: 12790: 12721: 12691: 12618: 12564: 12538: 12511: 12491: 12321: 12301: 12231: 12208: 12145: 12125: 12079: 12033: 12006: 11979: 11941: 11867: 11816: 11745: 11694: 11666: 11597: 11546: 11283: 11061: 10727: 10517: 10367: 10162: 10009: 9861: 9714: 9644: 9618: 8915: 8892: 8872: 8824: 8645: 8565: 8467: 8444: 8424: 8352: 8340:

is the quantile of the standard normal distribution.

8310: 8178: 8155: 8123: 8055: 8009: 7608: 7521: 7461: 7429: 6921: 6818: 6726: 6575: 6537: 6482: 6465:{\displaystyle \operatorname {GVar} =e^{\sigma ^{2}}} 6426: 6365: 6348:{\displaystyle \operatorname {GM} =e^{\mu }=\mu ^{*}} 6303: 6239: 6200: 6168: 6127: 6089: 6069: 6037: 6013: 5797: 5774: 5745: 5620: 5552: 5526: 5484: 5406: 5317: 5308:

All moments of the log-normal distribution exist and

5139: 5039: 5014: 4992: 4940: 4884: 4708: 4648: 4622: 4539: 3915: 3857: 3831: 3805: 3738: 3700: 3680: 3654: 3587: 3561: 3520: 3479: 3473:

can be used. They have a more direct interpretation:

3433: 3387: 3282: 3172: 3131: 3104: 3098:

In order to produce a distribution with desired mean

3069: 3033: 3000: 2959: 2914: 2869: 2826: 2785: 2752: 2712: 2692: 2647: 2618: 2598: 2578: 2548: 2258: 2097: 1966: 1806: 1695: 1550: 1430: 1276: 1169: 1053: 984: 932: 856: 785: 688: 526: 388: 327: 282: 221: 151: 112: 78: 52: 26784: 26546:

Douceur, John R.; Bolosky, William J. (1999-05-01).

26053:

Drainage Principles and Applications, Publication 16

25087:

Confidence Intervals for Risk Ratios and Odds Ratios

3902:

standard normal distribution, then we have that the

2467:

The distribution is occasionally referred to as the

26398: 24913: 24628: 21909: 21812:In operating-rooms management, the distribution of 19753:{\displaystyle {\hat {\mu }}_{1}-{\hat {\mu }}_{2}} 15272:is the density function of the normal distribution 14007:For a more accurate approximation, one can use the 11046: 8842:. The derivation of the formula is provided in the 6277:

Probabilities of functions of a log-normal variable

6263: 6162:is computed by transforming to the normal variable 4521: 264:{\displaystyle \ \mu \in (\ -\infty ,+\infty \ )\ } 26492: 25710:Scheler, Gabriele; Schumann, Johann (2006-10-08). 23918:Journal of the Royal Statistical Society, Series B 23600: 22968: 22875: 22733: 22697: 22541: 22396: 22101: 22072: 22013: 21993: 21942: 21681: 21638: 21618: 21586: 21551: 21521: 21015: 20533: 20214: 19816: 19784: 19752: 19687: 19201: 18941: 18739: 18712: 18685: 18634: 18571: 18519: 18468: 18254: 18117: 18073: 17912: 17799: 17688: 17660: 17592: 17563: 17497: 17314: 17163:The Cox Method proposes to plug-in the estimators 17148: 17117: 17043: 16962: 16910: 16858: 16797: 16762: 16604: 16485: 16435: 16361: 16172: 16152: 16132: 16100: 16057: 16028: 15966: 15922: 15902: 15882: 15856: 15693: 15613: 15593: 15573: 15546: 15516: 15310: 15264: 15244: 15089: 14949: 14905: 14854: 14800: 14744: 14682: 14637: 14611: 14557: 13996: 13730: 13697: 13670: 13348: 13328: 13308: 13254: 13234: 13214: 13141: 13077: 13024: 12954: 12882: 12836: 12816: 12796: 12746: 12704: 12677: 12604: 12550: 12524: 12497: 12464: 12307: 12287: 12214: 12194: 12131: 12111: 12065: 12019: 11992: 11953: 11927: 11853: 11800: 11731: 11678: 11652: 11583: 11521:{\textstyle v=\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})} 11520: 11444: 11345: 11266: 11182: 11123: 11033: 10691: 10485: 10347: 10138: 9985: 9841: 9670: 9630: 9596: 8898: 8878: 8830: 8807: 8628: 8544: 8450: 8430: 8402: 8332: 8293: 8161: 8141: 8102: 8038: 7945: 7572: 7504: 7447: 7409: 6895: 6759: 6709: 6543: 6527:, has been proposed. This term was intended to be 6520:{\displaystyle \operatorname {GCV} =e^{\sigma }-1} 6519: 6464: 6409: 6347: 6251: 6225: 6186: 6154: 6113: 6075: 6043: 6019: 5996: 5780: 5760: 5721: 5583: 5532: 5512: 5469: 5389: 5285: 5115: 5022: 5000: 4978: 4922: 4849: 4688: 4634: 4605: 4510: 3894: 3843: 3817: 3788: 3722: 3686: 3666: 3640: 3573: 3539: 3498: 3465: 3419: 3371: 3268: 3158: 3117: 3087: 3055: 3019: 2986: 2945: 2900: 2851: 2812: 2764: 2718: 2698: 2675: 2630: 2604: 2584: 2560: 2364: 2242: 2074: 1951: 1784: 1669: 1515: 1408: 1254: 1147: 1031: 962: 910: 834: 770: 666: 504: 366: 300: 263: 199: 130: 84: 64: 26717: 26699:Communications in Statistics - Theory and Methods 26669: 25714:. 36th Society for Neuroscience Meeting, Atlanta. 25326: 24870: 24560: 24399:Daly, Leslie E.; Bourke, Geoffrey Joseph (2000). 24238: 23990:Communications in Statistics - Theory and Methods 21034:in the ratio of the two expectations to create a 18583:Confidence interval for comparing two log normals 17009: 17008: 16728: 16727: 16583: 16582: 13911: 13774: 13579: 13393: 12452: 12371: 10954: 10871: 8759: 4308: 4201: 4147: 4146: 4068: 3987: 28144: 26631:Lognormal Distributions, Theory and Applications 25712:Diversity and stability in neuronal output rates 25466:"What is the average length of a game of chess?" 24875:. Las Vegas, NV, USA: IEEE. pp. 1880–1890. 24050:Barouch, E.; Kaufman, GM.; Glasser, ML. (1986). 22194:, is often modeled as a log-normal distribution. 21737:games tends to follow a log-normal distribution. 15526:Since the first term is constant with regard to 15311:{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} 9607: 8629:{\displaystyle g(k)=\operatorname {E} P(X>k)} 2638:. Then, the distribution of the random variable 1678: is asymptotically divergent, but adequate 835:{\displaystyle =\exp(\mu +\sigma \Phi ^{-1}(p))} 26261: 26081:CumFreq, free software for distribution fitting 25725:Mizuseki, Kenji; Buzsáki, György (2013-09-12). 25724: 25709: 25371:Buzsáki, György; Mizuseki, Kenji (2017-01-06). 25370: 24666: 21839:The concentration of rare elements in minerals. 17127: 12202:. This is easily generalized to the product of 7991:of two log-normal distributions with different 7512:. For a log-normal distribution it is equal to 3550: 28168:Infinitely divisible probability distributions 26545: 26216: 25183:Science China Physics, Mechanics and Astronomy 24684: 23920:, vol. 25, no. 2, pp. 392–393, 23739:National Institute of Standards and Technology 22985:The bias can be partially minimized by using: 22212:) follows a log-normal distribution over five 21692: 21587:{\displaystyle \sigma ={\frac {1}{\sqrt {6}}}} 18572:{\displaystyle t_{n-1,1-{\frac {\alpha }{2}}}} 16378:Reference range § Log-normal distribution 11534: 6760:{\displaystyle e^{-{\frac {1}{2}}\sigma ^{2}}} 6562:and is a consequence of the logarithm being a 6357:geometric or multiplicative standard deviation 5597:, but is not defined for any complex value of 26770: 26629:Crow, Edwin L.; Shimizu, Kunio, eds. (1988), 26548:"A large-scale study of file-system contents" 25578: 25447: 25037: 25035: 25033: 24678: 24561:Limpert, E.; Stahel, W. A.; Abbt, M. (2001). 24401:Interpretation and Uses of Medical Statistics 24207: 24205: 19760:follows a normal distribution, and that both 18118:{\displaystyle \mu +{\frac {\sigma ^{2}}{2}}} 16773: 15930:can be obtained by replacing the denominator 15318:. Therefore, the log-likelihood function is 15090:{\displaystyle F(x;\mu ,\sigma )=\left^{-1}.} 8418:The partial expectation of a random variable 4105: 4071: 4012: 3990: 26552:ACM SIGMETRICS Performance Evaluation Review 26445: 26141: 25535: 25295:Sutton, John (Mar 1997). "Gibrat's Legacy". 25176: 24934: 24491:Swat, MJ; Grenon, P; Wimalaratne, S (2016). 16399:: For the normal distribution, the interval 3906:of the log-normal distribution is given by: 2537: 2426:has a normal distribution. Equivalently, if 26628: 25101:Wu, Ziniu; Li, Juan; Bai, Chenyuan (2017). 24955: 24705: 23949: 23770:Continuous univariate distributions. Vol. 1 15110: 3895:{\displaystyle \ {\mathcal {N}}(\ 0,1\ )\ } 3466:{\displaystyle \ \sigma ^{*}=e^{\sigma }\ } 26777: 26763: 26446:Dexter, A. R.; Tanner, D. W. (July 1972). 26399:O'Connor, Patrick; Kleyner, Andre (2011). 26176: 26042: 25251: 25245: 25030: 24928: 24864: 24810:Asmussen, S.; Rojas-Nandayapa, L. (2008). 24398: 24202: 23983: 23981: 18686:{\displaystyle (\mu _{2},\sigma _{2}^{2})} 18635:{\displaystyle (\mu _{1},\sigma _{1}^{2})} 18520:{\displaystyle z_{1-{\frac {\alpha }{2}}}} 17600:is a normal distribution with parameters: 12678:{\displaystyle \sigma ^{2}={\mbox{var}}/n} 8861: 8555:Alternatively, by using the definition of 7970: 4873: 2746:the expectation and standard deviation of 1249: 26687: 26592: 26563: 26506: 26297: 26279: 26230: 26118: 25972: 25915: 25866: 25856: 25815: 25758: 25596: 25404: 25344: 25228: 25194: 25126: 24919: 24880: 24803: 24794: 24578: 24538: 24536: 24516: 24426: 24375: 24302: 24292: 24256: 24185: 24167: 24114: 24078: 24017:Journal of the Optical Society of America 23925: 23888: 23824: 23649: 22258:Modified lognormal power-law distribution 21682:{\textstyle \sigma =1{\big /}{\sqrt {6}}} 21030:It's worth noting that naively using the 16036:are not available, but the sample's mean 16029:{\displaystyle x_{1},x_{2},\ldots ,x_{n}} 15534:, both logarithmic likelihood functions, 14341: 14218: 8755: 8710: 8532: 6777:Itō's lemma for geometric Brownian motion 6775:, this is the same correction term as in 4911: 4067: 4054: 3986: 3973: 2023: 1892: 1831: 367:{\displaystyle \ x\in (\ 0,+\infty \ )\ } 175: 26724:Advances in Futures and Options Research 26662:Aitchison, J. and Brown, J.A.C. (1957) 26320: 24211: 24123: 23871:. ISI Proceedings: 53rd Session. Seoul. 23803: 21845: 15967:{\displaystyle {\widehat {\sigma }}^{2}} 15105: 12904: 12485:The geometric or multiplicative mean of 9688: 7974: 6058: 5540:, since the defining integral diverges. 3027:is log-normally distributed, then so is 2774: 2515:maximum entropy probability distribution 26696: 26670:Limpert, E; Stahel, W; Abbt, M (2001). 26207:, University of Sydney coursebook, 2007 25838: 25669:Journal of Biopharmaceutical Statistics 25100: 25058:https://stats.stackexchange.com/q/33395 24961: 24907: 24726:10.1061/(ASCE)0733-9429(1990)116:7(946) 24711: 24239:Limpert, E; Stahel, W; Abbt, M (2001). 24093: 24087: 24052:"On sums of lognormal random variables" 24043: 24014: 24008: 23987: 23978: 23861: 21744: 16390: 10761: 10757: 10746: 10742: 10544: 10540: 10531: 10389: 10381: 10187: 10184: 10176: 10031: 10023: 9883: 9875: 9736: 9728: 9612:In addition to the characterization by 8840:normal cumulative distribution function 6803:of a log-normally distributed variable 5150: 5050: 5016: 4994: 4904: 4886: 4699:This may also be expressed as follows: 1534: non-positive real part, see text 14: 28145: 26262:Sheridan, Paul; Onodera, Taku (2020). 26108: 26045:"6: Frequency and Regression Analysis" 25995: 25489: 25454:. ACM International Conference on KDD. 25294: 25096: 25094: 25009: 24843: 24837: 24772: 24770: 24738: 24593: 24533: 24349: 24274: 24145: 24143: 24141: 22969:{\displaystyle E=e^{log(p_{1})}=p_{1}} 12112:{\displaystyle \mu =\mu _{1}-\mu _{2}} 12066:{\displaystyle \mu =\mu _{1}+\mu _{2}} 8413: 2513:). The log-normal distribution is the 1532: defined only for numbers with a 26758: 26403:. John Wiley & Sons. p. 35. 26359: 25432: 25428: 25426: 25424: 25366: 25364: 25322: 25320: 25318: 25024:10.1061/(ASCE)1084-0699(2002)7:6(441) 23915: 23804:Park, Sung Y.; Bera, Anil K. (2009). 23763: 23761: 23759: 23757: 23755: 23701: 23620: 21895:The rainfall data are represented by 16371: 12883:{\displaystyle H={\frac {G^{2}}{A}}.} 10150:LogNormal4(m,cv) with median, m, and 9682:, the knowledge base and ontology of 6782: 3420:{\displaystyle \ \mu ^{*}=e^{\mu }\ } 28127: 25170: 23909: 23865:Multivariate Log–Normal Distribution 23819:(2): 219–230, esp. Table 1, p. 221. 23725: 23723: 23697: 23695: 23693: 23691: 22014:{\displaystyle \operatorname {erf} } 21700:Multiplicative Central Limit Theorem 16133:{\displaystyle \operatorname {Var} } 15581:, reach their maximum with the same 12475:Multiplicative central limit theorem 11183:{\textstyle \mu _{N}=\exp(\mu +v/2)} 9671:{\displaystyle \mu ^{*},\sigma ^{*}} 5736:representations have been obtained. 3159:{\displaystyle \ \sigma _{X}^{2}\ ,} 2908:is normally distributed, then so is 2521:—for which the mean and variance of 2509:in the log domain (sometimes called 2430:has a normal distribution, then the 25144:Probabilistic Engineering Mechanics 25091: 24776: 24767: 24149: 24138: 23731:"1.3.6.6.9. Lognormal Distribution" 13731:{\displaystyle \sigma _{j}=\sigma } 13309:{\textstyle Y=\sum _{j=1}^{n}X_{j}} 7448:{\displaystyle \operatorname {CV} } 6544:{\displaystyle \operatorname {CV} } 6289:Geometric or multiplicative moments 3581:is log-normally distributed (i.e., 2813:{\displaystyle \ Y=\mu +\sigma Z\ } 2676:{\displaystyle X=e^{\mu +\sigma Z}} 24: 26622: 25421: 25361: 25315: 25141: 25135: 24858:10.1002/j.1538-7305.1967.tb04244.x 24819:Statistics and Probability Letters 23752: 22087: 22041: 21874:The image on the right, made with 17160:as well as various other methods. 16101:{\displaystyle \operatorname {E} } 16083: 15281: 14721: 14697: 14674: 14293: 14271: 14091: 14042: 13112: 13085:is distributed log-normally, then 12925: 12901:, which can be easily drawn from. 12545: 9539: 9488: 9425: 9361: 9220: 9158: 9043: 8987: 8825: 8756: 8671: 8581: 8493: 8333:{\displaystyle q_{\Phi }(\alpha )} 8316: 8272: 8220: 7906: 7837: 7813: 7747: 7705: 7660: 7636: 7483: 7293: 7130: 7102: 7077: 6990: 6926: 6819: 6576: 6474:geometric coefficient of variation 6297:of the log-normal distribution is 6114:{\displaystyle \mu =1,\sigma =0.5} 5637: 5584:{\displaystyle \operatorname {E} } 5553: 5513:{\displaystyle \operatorname {E} } 5485: 5318: 5257: 5231: 5215: 5096: 5040: 4895: 4654: 4626: 4562: 4244: 4237: 4142: 4130: 4123: 4060: 4037: 4030: 3979: 3956: 3949: 3863: 3809: 3759: 3674:is normally distributed with mean 2413:is log-normally distributed, then 2329: 2322: 1680: for most numerical purposes 1570: 963:{\displaystyle \ \exp(\ \mu \ )\ } 808: 623: 352: 249: 240: 25: 28184: 26743: 26401:Practical Reliability Engineering 25012:Journal of Hydrologic Engineering 23892:The Lognormal Random Multivariate 23720: 23688: 22741:was log-normal, we'd expect that 21721: 17593:{\displaystyle {\widehat {\mu }}} 16963:{\displaystyle \mu ^{*}=e^{\mu }} 16781:Using the principle, note that a 16382:The most efficient way to obtain 14683:{\displaystyle x\in (c,+\infty )} 13705:have the same variance parameter 11540:Multiplication by a constant: If 5732:However, a number of alternative 5023:{\displaystyle {\boldsymbol {Y}}} 5001:{\displaystyle {\boldsymbol {X}}} 4979:{\displaystyle Y_{i}=\exp(X_{i})} 28163:Exponential family distributions 28126: 28117: 28116: 25839:Scheler, Gabriele (2017-07-28). 24714:Journal of Hydraulic Engineering 24350:Schiff, MH; et al. (2014). 24094:Leipnik, Roy B. (January 1991). 23905:from the original on 2015-09-30. 23878:from the original on 2013-07-19. 22883:. But in practice, we KNOW that 21910:Social sciences and demographics 17326:approximate confidence intervals 14968:to get an approximation for the 11047:Examples for re-parameterization 10154:, cv, both on the natural scale 6295:geometric or multiplicative mean 6264:Probability in different domains 4932:multivariate normal distribution 4913: 4528:cumulative distribution function 4522:Cumulative distribution function 3723:{\displaystyle \ \sigma ^{2}\ :} 2946:{\displaystyle \ \log _{b}(X)\ } 2901:{\displaystyle \ \log _{a}(X)\ } 301:{\displaystyle \ \sigma >0\ } 100: 98:Cumulative distribution function 39: 26610: 26601: 26580: 26539: 26495:The European Physical Journal B 26486: 26439: 26417: 26392: 26353: 26323:"Gibrat's Law for (All) Cities" 26314: 26255: 26210: 26197: 26170: 26135: 26102: 26085: 26074: 26036: 26002:Geochimica et Cosmochimica Acta 25989: 25932: 25883: 25832: 25800:10.1016/j.pneurobio.2012.09.004 25775: 25718: 25703: 25656: 25621: 25572: 25529: 25516: 25483: 25458: 25441: 25288: 25274:10.1016/j.actaastro.2018.11.033 25080: 25071: 25062: 25050: 25003: 24657: 24622: 24587: 24554: 24545: 24484: 24463: 24392: 24368:10.1136/annrheumdis-2014-205228 24343: 24319: 24232: 23677:from the original on 2021-04-18 22979: 22287: 22278: 17223: 16275: 15761: 6787:For any real or complex number 6226:{\displaystyle \sin e^{x}>0} 5400:This can be derived by letting 3648:), if the natural logarithm of 3540:{\displaystyle \ \sigma ^{*}\ } 2852:{\displaystyle \ X\sim e^{Y}\ } 2409:. Thus, if the random variable 26180:The (mis-)Behaviour of Markets 25858:10.12688/f1000research.12130.2 25297:Journal of Economic Literature 24753:10.1080/03461238.1977.10405635 24741:Scandinavian Actuarial Journal 24596:Comput Methods Programs Biomed 24059:Studies in Applied Mathematics 23943: 23882: 23855: 23797: 23589: 23570: 23540: 23521: 23421: 23385: 23369: 23325: 23295: 23276: 23246: 23227: 23120: 23107: 23090: 23077: 23039: 23026: 23018: 23005: 22948: 22935: 22915: 22903: 22893: 22868: 22849: 22841: 22826: 22806: 22793: 22773: 22761: 22751: 22734:{\displaystyle {\hat {p}}_{1}} 22719: 22690: 22687: 22669: 22659: 22651: 22639: 22623: 22617: 22605: 22595: 22575: 22562: 22536: 22533: 22514: 22506: 22491: 22485: 22472: 22460: 22442: 22430: 22420: 22391: 22380: 22365: 22339: 22312: 22096: 22090: 21753:Incubation period of diseases. 21619:{\displaystyle e=2.718\ldots } 21506: 21487: 21457: 21438: 21332: 21296: 21280: 21254: 21237: 21201: 21185: 21141: 21116: 21103: 21086: 21073: 20998: 20979: 20949: 20930: 20814: 20746: 20724: 20714: 20599: 20586: 20578: 20565: 20523: 20504: 20474: 20455: 20339: 20271: 20249: 20239: 20201: 20182: 20152: 20133: 20042: 20006: 19990: 19964: 19950: 19882: 19860: 19850: 19738: 19716: 19670: 19651: 19621: 19602: 19486: 19418: 19396: 19386: 19271: 19258: 19250: 19237: 19194: 19158: 19142: 19116: 19011: 18998: 18990: 18977: 18843: 18821: 18798: 18765: 18680: 18649: 18629: 18598: 18451: 18439: 18349: 18289: 18283: 18244: 18232: 18142: 18060: 18048: 17899: 17887: 17525: 17519: 17480: 17468: 17378: 17356: 17353: 17347: 17341: 17094: 17071: 17038: 17029: 17015: 16977: 16853: 16812: 16757: 16748: 16734: 16705: 16699: 16690: 16676: 16651: 16637: 16619: 16599: 16560: 16554: 16500: 16480: 16450: 16430: 16406: 16339: 16253: 16210: 16127: 16121: 16095: 16089: 16049: 15688: 15508: 15433: 15385: 15328: 15305: 15286: 15236: 15217: 15157: 15145: 15059: 15046: 14999: 14981: 14944: 14932: 14900: 14881: 14849: 14843: 14795: 14789: 14777: 14765: 14733: 14727: 14715: 14703: 14677: 14662: 14606: 14587: 14312: 14299: 14290: 14277: 14110: 14097: 13861: 13837: 13806: 13780: 13529: 13479: 13474: 13443: 13209: 13178: 13136: 13117: 13104: 13098: 13072: 13053: 13016: 12997: 12985: 12979: 12949: 12930: 12741: 12728: 12664: 12661: 12648: 12639: 12599: 12596: 12583: 12574: 12542: 12282: 12251: 11922: 11887: 11848: 11829: 11792: 11767: 11726: 11707: 11647: 11613: 11578: 11559: 11515: 11471: 11340: 11328: 11319: 11316: 11290: 11253: 11247: 11236: 11216: 11177: 11157: 11118: 11092: 11083: 11080: 11068: 11020: 10976: 10767: 10731: 10678: 10665: 10644: 10630: 10584: 10571: 10550: 10521: 10469: 10450: 10393: 10371: 10334: 10312: 10298: 10284: 10236: 10214: 10191: 10166: 10110: 10096: 10035: 10013: 9958: 9939: 9887: 9865: 9807: 9788: 9740: 9718: 9568: 9555: 9517: 9504: 9454: 9441: 9390: 9377: 9315: 9312: 9286: 9271: 9242: 9236: 9199: 9193: 9112: 9094: 9065: 9059: 9009: 9003: 8941: 8923: 8707: 8689: 8655: 8649: 8623: 8611: 8605: 8587: 8575: 8569: 8529: 8511: 8477: 8471: 8365: 8359: 8327: 8321: 8283: 8277: 8263: 8249: 8231: 8225: 8195: 8189: 8068: 8062: 8023: 8010: 7919: 7912: 7901: 7895: 7850: 7843: 7832: 7819: 7760: 7753: 7741: 7735: 7718: 7711: 7679: 7666: 7649: 7642: 7534: 7528: 7495: 7489: 7478: 7472: 7442: 7436: 7305: 7299: 7285: 7279: 7261: 7255: 7239: 7213: 7181: 7155: 7146: 7142: 7136: 7127: 7115: 7108: 7096: 7083: 7067: 7061: 7009: 6996: 6938: 6932: 6838: 6825: 6767:is sometimes interpreted as a 6699: 6693: 6679: 6673: 6588: 6582: 6495: 6489: 6439: 6433: 6378: 6372: 6316: 6310: 6155:{\displaystyle p(\sin y>0)} 6149: 6131: 6083:is a log-normal variable with 5988: 5956: 5919: 5887: 5875: 5843: 5807: 5801: 5755: 5749: 5655: 5645: 5591:is defined for real values of 5578: 5559: 5507: 5491: 5457: 5435: 5429: 5423: 5337: 5324: 5277: 5248: 5243: 5211: 5155: 5146: 5055: 5046: 4973: 4960: 4917: 4900: 4680: 4662: 4584: 4572: 4556: 4550: 4460: 4441: 3936: 3930: 3886: 3868: 3783: 3764: 3751: 3745: 2937: 2931: 2892: 2886: 2820:is normally distributed, then 2532: 2359: 2356: 2347: 2341: 2325: 2313: 2223: 2208: 2038: 2031: 2007: 2001: 1907: 1900: 1876: 1870: 1845: 1839: 1594: 1581: 1240: 1227: 1084: 1071: 954: 942: 829: 826: 820: 795: 757: 742: 645: 639: 358: 337: 255: 231: 13: 1: 26718:Brooks, Robert; Corson, Jon; 26666:, Cambridge University Press. 25598:10.1093/bioinformatics/bts649 25164:10.1016/S0266-8920(03)00028-6 24846:Bell System Technical Journal 24643:10.1016/S0169-2607(00)00117-6 24509:10.1093/bioinformatics/btw170 24325:Sawant, S.; Mohan, N. (2011) 23950:Billingsley, Patrick (2012). 23835:10.1016/j.jeconom.2008.12.014 23637:Annals of Operations Research 23614: 22173: 21919:, there is evidence that the 21901:cumulative frequency analysis 20224:Based on the above, standard 18081:Based on the above, standard 15890:is unbiased, but the one for 13738:, these formulas simplify to 12893:Log-normal distributions are 9608:Alternative parameterizations 8886:—with respect to a threshold 6054: 2987:{\displaystyle \ a,b\neq 1~.} 2953:for any two positive numbers 26429:www.WirelessCommunication.NL 26144:Journal of Political Economy 26022:10.1016/0016-7037(54)90040-X 25996:Ahrens, L. H. (1954-02-01). 25743:10.1016/j.celrep.2013.07.039 25642:10.1016/0021-9681(79)90070-5 25377:Nature Reviews. Neuroscience 24453:print edition. Online eBook 24275:Heil P, Friedrich B (2017). 23644:(1–2). Springer: 1281–1315. 22253:Log-distance path loss model 21857: 21823: 20228:can be constructed (using a 18125:can be constructed (using a 17128:Confidence interval for E(X) 16395:A basic example is given by 13149:is a normal random variable. 12779:Lorenz asymmetry coefficient 12551:{\displaystyle n\to \infty } 10719:, both on the natural scale 10509:, both on the natural scale 10503:geometric standard deviation 9706:, σ, both on the log-scale 9631:{\displaystyle \mu ,\sigma } 8438:with respect to a threshold 6771:. From the point of view of 6355:. It equals the median. The 4868:complementary error function 3904:probability density function 3844:{\displaystyle \ \varphi \ } 3551:Probability density function 3499:{\displaystyle \ \mu ^{*}\ } 3088:{\displaystyle 0<a\neq 1} 2859:is log-normally distributed. 2631:{\displaystyle \sigma >0} 37:Probability density function 7: 26177:Mandelbrot, Benoit (2004). 25630:Journal of Chronic Diseases 25524:American journal of hygiene 25492:Problems of relative growth 23927:10.1007/978-1-4419-5823-5_6 23684:– via stonybrook.edu. 22241: 21830:Particle size distributions 21702:". This is also known as 21693:Occurrence and applications 16922:is the 97.5% quantile of a 15977:When the individual values 12705:{\displaystyle \sigma ^{2}} 11535:Multiple, reciprocal, power 10715:, and standard deviation, σ 6912:are respectively given by: 5761:{\displaystyle \varphi (t)} 3555:A positive random variable 10: 28189: 27950:Wrapped asymmetric Laplace 26921:Extended negative binomial 26664:The Lognormal Distribution 26525:10.1140/epjb/e2011-20581-3 26290:10.1038/s41598-018-21133-2 26050:. In Ritzema, H.P. (ed.). 25490:Huxley, Julian S. (1932). 24608:10.1016/j.cmpb.2012.05.005 23660:10.1007/s10479-019-03373-1 19832:normally distributed (via 17704:normally distributed (via 17324:and use them to construct 16918:is the standard error and 16375: 16058:{\displaystyle {\bar {x}}} 14649:distribution with support 14647:Three-parameter log-normal 12761:This is commonly known as 12747:{\displaystyle \ln(X_{i})} 12478: 8039:{\displaystyle (\ln f)'=0} 3056:{\displaystyle \ a^{Y}\ ,} 2738:of the variable's natural 2612:be two real numbers, with 131:{\displaystyle \ \mu =0\ } 28112: 28046: 28004: 27905: 27741: 27719: 27710: 27609:Generalized extreme value 27594: 27429: 27389:Relativistic Breit–Wigner 27105: 27002: 26993: 26886: 26806: 26797: 26786:Probability distributions 26711:10.1080/03610928908830173 26241:10.1016/j.joi.2014.09.011 26129:10.1007/978-4-431-66993-7 26043:Oosterbaan, R.J. (1994). 25681:10.1080/10543409708835177 25213:10.1007/s11433-013-5321-0 24831:10.1016/j.spl.2008.03.035 24116:10.1017/S0334270000006901 24002:10.1080/03610928908830173 23889:Halliwell, Leigh (2015). 23704:"Log Normal Distribution" 22248:Heavy-tailed distribution 22225:In computer networks and 21800:elimination rate constant 19817:{\displaystyle S_{2}^{2}} 19785:{\displaystyle S_{1}^{2}} 15574:{\displaystyle \ell _{N}} 15099:log-logistic distribution 14813:Johnson's SU-distribution 9684:probability distributions 4635:{\displaystyle \ \Phi \ } 3818:{\displaystyle \ \Phi \ } 3514:of the distribution, and 3020:{\displaystyle \ e^{Y}\ } 2538:Generation and parameters 2090: 2085: 1800: 1795: 1689: 1684: 1543: 1538: 1531: 1526: 1424: 1419: 1270: 1265: 1163: 1158: 1047: 1042: 978: 973: 926: 921: 850: 845: 682: 677: 520: 515: 382: 377: 321: 316: 215: 210: 145: 142: 96: 72:but differing parameters 35: 28153:Continuous distributions 26339:10.1257/0002828043052303 26327:American Economic Review 25965:10.1088/1367-2630/aac3c7 25788:Progress in Neurobiology 24937:Water Resources Research 24891:10.1109/WSC.2017.8247924 24294:10.3389/fpsyg.2016.02013 23862:Tarmast, Ghasem (2001). 22271: 22102:{\displaystyle \Phi (x)} 21834:molar mass distributions 19826:chi-squared distribution 17698:chi-squared distribution 16774:Confidence interval for 15265:{\displaystyle \varphi } 15111:Estimation of parameters 12768: 11954:{\displaystyle a\neq 0.} 11353:following formulas hold 11131:following formulas hold 10152:coefficient of variation 9686:lists seven such forms: 7422:coefficient of variation 6553:Coefficient of variation 6044:{\displaystyle \varphi } 3118:{\displaystyle \mu _{X}} 2570:standard normal variable 2395:probability distribution 65:{\displaystyle \ \mu \ } 27:Probability distribution 27604:Generalized chi-squared 27548:Normal-inverse Gaussian 26452:Nature Physical Science 26219:Journal of Informetrics 26205:Advanced Option Pricing 25470:chess.stackexchange.com 24949:10.1029/WR006i002p00505 24672:ProbOnto website, URL: 24281:Frontiers in Psychology 23953:Probability and Measure 23813:Journal of Econometrics 21943:{\displaystyle \sigma } 21639:{\displaystyle \sigma } 21552:{\displaystyle \sigma } 16173:{\displaystyle \sigma } 15938:−1 in the equation for 15923:{\displaystyle \sigma } 15903:{\displaystyle \sigma } 15614:{\displaystyle \sigma } 13316:. The distribution of 13235:{\displaystyle \sigma } 12132:{\displaystyle \sigma } 11973:, log-normal variables 11679:{\displaystyle a>0.} 8862:Conditional expectation 8559:, it can be written as 8557:conditional expectation 8142:{\displaystyle Y=\ln X} 7971:Mode, median, quantiles 6187:{\displaystyle x=\ln y} 5545:characteristic function 5298:univariate distribution 4874:Multivariate log-normal 2719:{\displaystyle \sigma } 2605:{\displaystyle \sigma } 85:{\displaystyle \sigma } 31:Log-normal distribution 28173:Non-Newtonian calculus 27916:Univariate (circular) 27477:Generalized hyperbolic 26906:Conway–Maxwell–Poisson 26896:Beta negative binomial 26472:10.1038/physci238031a0 26366:Statistics in Medicine 26321:Eeckhout, Jan (2004). 25945:New Journal of Physics 25908:10.1093/cercor/bhad160 25538:Statistics in Medicine 24332:24 August 2011 at the 24150:Das, Abhranil (2021). 24037:10.1364/JOSA.66.000211 23602: 22970: 22877: 22735: 22699: 22543: 22398: 22188:wireless communication 22131:log-Lévy distributions 22103: 22074: 22015: 21995: 21944: 21854: 21683: 21640: 21620: 21588: 21553: 21523: 21017: 20535: 20216: 19818: 19786: 19754: 19689: 19203: 18943: 18741: 18714: 18687: 18636: 18573: 18521: 18470: 18256: 18119: 18075: 17914: 17801: 17690: 17662: 17594: 17565: 17499: 17328:in the following way: 17316: 17150: 17119: 17045: 16964: 16912: 16860: 16799: 16764: 16606: 16487: 16437: 16363: 16174: 16154: 16134: 16102: 16059: 16030: 15968: 15924: 15904: 15884: 15858: 15695: 15615: 15595: 15575: 15548: 15518: 15312: 15266: 15246: 15183: 15091: 14951: 14907: 14856: 14802: 14746: 14684: 14639: 14613: 14559: 13998: 13732: 13699: 13672: 13350: 13330: 13310: 13295: 13256: 13236: 13216: 13143: 13079: 13026: 12956: 12884: 12838: 12818: 12798: 12748: 12706: 12679: 12606: 12605:{\displaystyle \mu =E} 12552: 12526: 12499: 12466: 12434: 12397: 12349: 12309: 12289: 12216: 12196: 12133: 12113: 12067: 12021: 11994: 11955: 11929: 11855: 11802: 11733: 11680: 11654: 11585: 11522: 11446: 11347: 11268: 11184: 11125: 11035: 10693: 10501:) with median, m, and 10487: 10349: 10140: 9987: 9843: 9694: 9672: 9632: 9598: 8900: 8880: 8832: 8809: 8630: 8546: 8452: 8432: 8404: 8334: 8295: 8163: 8143: 8104: 8040: 7996: 7947: 7574: 7506: 7449: 7411: 6897: 6761: 6711: 6545: 6521: 6466: 6411: 6349: 6260: 6253: 6252:{\displaystyle \sin y} 6227: 6188: 6156: 6115: 6077: 6045: 6021: 5998: 5782: 5762: 5723: 5641: 5585: 5534: 5514: 5471: 5391: 5287: 5117: 5024: 5002: 4980: 4924: 4851: 4690: 4636: 4607: 4512: 3896: 3845: 3819: 3790: 3724: 3688: 3668: 3642: 3575: 3541: 3500: 3467: 3421: 3373: 3270: 3160: 3119: 3089: 3057: 3021: 2988: 2947: 2902: 2860: 2853: 2814: 2766: 2720: 2700: 2677: 2632: 2606: 2586: 2562: 2366: 2244: 2076: 1953: 1786: 1671: 1574: 1544: representation 1517: 1410: 1256: 1149: 1033: 964: 912: 836: 772: 668: 506: 368: 302: 265: 201: 132: 86: 66: 27961:Bivariate (spherical) 27459:Kaniadakis κ-Gaussian 26565:10.1145/301464.301480 26360:Kault, David (1996). 24409:10.1002/9780470696750 24134:Thiele report 6 (13). 24071:10.1002/sapm198675137 23774:John Wiley & Sons 23708:mathworld.wolfram.com 23603: 22971: 22878: 22736: 22700: 22544: 22399: 22164:central limit theorem 22147:central limit theorem 22104: 22075: 22016: 21996: 21945: 21884:binomial distribution 21849: 21796:elimination half-life 21684: 21641: 21621: 21589: 21554: 21524: 21018: 20536: 20217: 19819: 19787: 19755: 19690: 19204: 18944: 18742: 18740:{\displaystyle n_{2}} 18715: 18713:{\displaystyle n_{1}} 18688: 18637: 18574: 18522: 18471: 18257: 18120: 18076: 17915: 17802: 17691: 17689:{\displaystyle S^{2}} 17663: 17595: 17566: 17500: 17317: 17151: 17120: 17046: 16965: 16913: 16861: 16800: 16765: 16607: 16488: 16438: 16376:Further information: 16364: 16175: 16155: 16135: 16103: 16060: 16031: 15969: 15925: 15905: 15885: 15859: 15696: 15616: 15596: 15576: 15549: 15547:{\displaystyle \ell } 15519: 15313: 15267: 15247: 15163: 15106:Statistical inference 15092: 14966:logistic distribution 14952: 14908: 14857: 14803: 14747: 14685: 14640: 14614: 14560: 13999: 13733: 13700: 13698:{\displaystyle X_{j}} 13678:In the case that all 13673: 13351: 13331: 13311: 13275: 13257: 13237: 13217: 13144: 13080: 13027: 12957: 12905:Related distributions 12885: 12839: 12819: 12799: 12756:central limit theorem 12749: 12707: 12680: 12607: 12553: 12527: 12525:{\displaystyle X_{i}} 12500: 12467: 12414: 12377: 12329: 12310: 12290: 12217: 12197: 12134: 12114: 12068: 12022: 12020:{\displaystyle X_{2}} 11995: 11993:{\displaystyle X_{1}} 11956: 11930: 11856: 11803: 11734: 11681: 11655: 11586: 11523: 11447: 11348: 11269: 11185: 11126: 11036: 10694: 10488: 10350: 10141: 9997:LogNormal3(m,σ) with 9988: 9844: 9698:LogNormal1(μ,σ) with 9692: 9673: 9633: 9599: 8901: 8881: 8856:Black–Scholes formula 8833: 8831:{\displaystyle \Phi } 8810: 8631: 8547: 8453: 8433: 8405: 8335: 8296: 8164: 8144: 8105: 8041: 7978: 7948: 7575: 7507: 7450: 7412: 6898: 6762: 6720:In finance, the term 6712: 6546: 6522: 6467: 6412: 6350: 6254: 6228: 6189: 6157: 6116: 6078: 6062: 6046: 6022: 5999: 5783: 5763: 5724: 5621: 5586: 5535: 5515: 5472: 5392: 5288: 5118: 5025: 5003: 4981: 4925: 4852: 4691: 4637: 4608: 4513: 3897: 3846: 3820: 3791: 3725: 3689: 3669: 3667:{\displaystyle \ X\ } 3643: 3576: 3574:{\displaystyle \ X\ } 3542: 3501: 3468: 3422: 3374: 3271: 3161: 3120: 3090: 3058: 3022: 2989: 2948: 2903: 2854: 2815: 2778: 2767: 2765:{\displaystyle \ X\ } 2721: 2701: 2678: 2633: 2607: 2587: 2563: 2561:{\displaystyle \ Z\ } 2517:for a random variate 2507:central limit theorem 2473:Galton's distribution 2456:sciences, as well as 2367: 2245: 2077: 1954: 1787: 1672: 1554: 1518: 1411: 1257: 1150: 1034: 965: 913: 837: 773: 669: 507: 369: 303: 266: 202: 133: 87: 67: 28026:Dirac delta function 27973:Bivariate (toroidal) 27930:Univariate von Mises 27801:Multivariate Laplace 27693:Shifted log-logistic 27042:Continuous Bernoulli 26435:on January 13, 2012. 24631:Comp Meth Pro Biomed 22989: 22887: 22745: 22709: 22553: 22408: 22302: 22139:stock market crashes 22116:, in particular the 22084: 22029: 22005: 21958: 21934: 21852:distribution fitting 21745:Biology and medicine 21653: 21630: 21601: 21563: 21543: 21048: 20545: 20236: 20226:confidence intervals 19847: 19836:, with the relevant 19796: 19764: 19706: 19217: 18968: 18756: 18724: 18697: 18693:, with sample sizes 18646: 18595: 18531: 18491: 18266: 18133: 18089: 18083:confidence intervals 17931: 17811: 17716: 17673: 17604: 17575: 17513: 17332: 17167: 17149:{\displaystyle \mu } 17140: 17055: 16974: 16934: 16870: 16809: 16798:{\displaystyle \mu } 16789: 16616: 16497: 16447: 16403: 16397:prediction intervals 16391:Prediction intervals 16184: 16164: 16153:{\displaystyle \mu } 16144: 16112: 16080: 16040: 15981: 15942: 15914: 15894: 15883:{\displaystyle \mu } 15874: 15870:, the estimator for 15705: 15625: 15605: 15594:{\displaystyle \mu } 15585: 15558: 15538: 15322: 15276: 15256: 15139: 15115:For determining the 14975: 14917: 14866: 14822: 14756: 14694: 14653: 14623: 14572: 14018: 13742: 13709: 13682: 13361: 13340: 13320: 13266: 13255:{\displaystyle \mu } 13246: 13226: 13156: 13089: 13038: 12970: 12914: 12899:stable distributions 12895:infinitely divisible 12851: 12828: 12808: 12788: 12719: 12689: 12616: 12562: 12536: 12509: 12489: 12319: 12299: 12229: 12206: 12143: 12123: 12077: 12031: 12004: 11977: 11939: 11865: 11814: 11743: 11692: 11664: 11595: 11544: 11456: 11357: 11281: 11194: 11135: 11059: 10725: 10515: 10365: 10160: 10007: 9859: 9712: 9642: 9616: 8913: 8890: 8870: 8822: 8643: 8563: 8465: 8442: 8422: 8350: 8308: 8176: 8153: 8121: 8053: 8007: 7606: 7519: 7459: 7427: 6919: 6816: 6769:convexity correction 6724: 6573: 6535: 6480: 6424: 6363: 6301: 6237: 6198: 6166: 6125: 6087: 6067: 6035: 6011: 5795: 5772: 5743: 5618: 5550: 5524: 5482: 5404: 5315: 5137: 5037: 5012: 4990: 4938: 4882: 4706: 4646: 4620: 4537: 3913: 3855: 3829: 3803: 3736: 3698: 3687:{\displaystyle \mu } 3678: 3652: 3585: 3559: 3518: 3477: 3431: 3385: 3280: 3170: 3129: 3102: 3067: 3031: 2998: 2957: 2912: 2867: 2824: 2783: 2750: 2710: 2699:{\displaystyle \mu } 2690: 2645: 2616: 2596: 2585:{\displaystyle \mu } 2576: 2546: 2432:exponential function 2407:normally distributed 2256: 2095: 1964: 1804: 1693: 1548: 1428: 1274: 1167: 1051: 982: 930: 854: 783: 686: 524: 386: 325: 280: 219: 149: 110: 76: 50: 46:Identical parameter 28158:Normal distribution 28074:Natural exponential 27979:Bivariate von Mises 27945:Wrapped exponential 27811:Multivariate stable 27806:Multivariate normal 27127:Benktander 2nd kind 27122:Benktander 1st kind 26911:Discrete phase-type 26517:2012EPJB...85...28G 26464:1972NPhS..238...31D 26014:1954GeCoA...5...49A 25957:2018NJPh...20e3057P 25526:51 (1950): 310-318. 25266:2019AcAau.155..118B 25205:2013SCPMA..56.2143W 25156:2003PEngM..18..241W 25119:2017Entrp..19...56W 24796:10.1155/2009/630857 24674:http://probonto.org 24178:10.1167/jov.21.10.1 24029:1976JOSA...66..211B 23702:Weisstein, Eric W. 23565: 23516: 23484: 23452: 23420: 23402: 23271: 23222: 23190: 23158: 22214:orders of magnitude 22118:Black–Scholes model 21925:Pareto distribution 21787:variables, such as 21482: 21433: 21401: 21369: 21331: 21313: 21236: 21218: 20974: 20925: 20893: 20861: 20813: 20785: 20686: 20644: 20499: 20450: 20418: 20386: 20338: 20310: 20177: 20128: 20096: 20064: 20041: 20023: 19949: 19921: 19813: 19781: 19646: 19597: 19565: 19533: 19485: 19457: 19358: 19316: 19193: 19175: 19098: 19056: 18924: 18899: 18679: 18628: 17134:confidence interval 16783:confidence interval 15133:normal distribution 14959:Suzuki distribution 14638:{\displaystyle X+c} 14542: 14469: 14426: 14340: 14217: 14187: 14158: 13981: 13763: 13655: 13620: 13517: 13465: 13440: 13382: 13208: 12964:normal distribution 12897:, but they are not 12449: 12281: 12225:More generally, if 12191: 12173: 11514: 11494: 11434: 11414: 11277:For the transition 11055:For the transition 11019: 10999: 10947: 10927: 10841: 10821: 8675: 8497: 8414:Partial expectation 6773:stochastic calculus 3355: 3336: 3247: 3229: 3208: 3149: 2469:Galton distribution 32: 27729:Rectified Gaussian 27614:Generalized Pareto 27472:Generalized normal 27344:Matrix-exponential 26349:– via JSTOR. 26268:Scientific Reports 23598: 23551: 23502: 23470: 23438: 23406: 23388: 23257: 23208: 23176: 23144: 22966: 22873: 22731: 22695: 22539: 22394: 22133:, which possesses 22129:have argued that 22099: 22070: 22011: 21991: 21940: 21897:plotting positions 21855: 21762:entropy production 21679: 21636: 21616: 21584: 21549: 21519: 21468: 21419: 21387: 21355: 21317: 21299: 21222: 21204: 21013: 20960: 20911: 20879: 20847: 20799: 20771: 20672: 20630: 20531: 20485: 20436: 20404: 20372: 20324: 20296: 20212: 20163: 20114: 20082: 20050: 20027: 20009: 19935: 19907: 19814: 19799: 19782: 19767: 19750: 19685: 19632: 19583: 19551: 19519: 19471: 19443: 19344: 19302: 19199: 19179: 19161: 19084: 19042: 18939: 18910: 18885: 18737: 18710: 18683: 18665: 18632: 18614: 18569: 18517: 18466: 18252: 18115: 18071: 17910: 17797: 17686: 17658: 17590: 17561: 17495: 17312: 17249: 17197: 17146: 17115: 17041: 16970:(the median), is: 16960: 16908: 16856: 16795: 16760: 16612:contains 2/3, and 16602: 16483: 16433: 16384:interval estimates 16372:Interval estimates 16359: 16170: 16150: 16130: 16098: 16067:standard deviation 16055: 16026: 15964: 15920: 15900: 15880: 15854: 15796: 15735: 15691: 15611: 15591: 15571: 15544: 15514: 15403: 15308: 15262: 15242: 15117:maximum likelihood 15087: 14947: 14903: 14852: 14798: 14742: 14680: 14645:is said to have a 14635: 14609: 14555: 14553: 14528: 14455: 14412: 14357: 14326: 14234: 14203: 14173: 14144: 14125: 14090: 14062: 14009:Monte Carlo method 13994: 13992: 13967: 13749: 13728: 13695: 13668: 13666: 13641: 13606: 13503: 13451: 13426: 13368: 13346: 13326: 13306: 13252: 13232: 13212: 13194: 13139: 13075: 13022: 12952: 12880: 12834: 12814: 12794: 12744: 12702: 12675: 12637: 12602: 12548: 12522: 12495: 12462: 12461: 12460: 12435: 12305: 12285: 12267: 12212: 12192: 12177: 12159: 12129: 12109: 12063: 12017: 11990: 11951: 11925: 11851: 11798: 11756: 11729: 11676: 11650: 11581: 11518: 11500: 11480: 11442: 11420: 11400: 11343: 11264: 11180: 11121: 11031: 11005: 10985: 10933: 10913: 10827: 10807: 10689: 10483: 10345: 10136: 9983: 9839: 9704:standard deviation 9695: 9668: 9628: 9594: 9592: 8896: 8876: 8828: 8805: 8741: 8661: 8626: 8542: 8483: 8448: 8428: 8400: 8330: 8291: 8159: 8139: 8100: 8036: 7997: 7943: 7941: 7570: 7502: 7500: 7445: 7407: 7405: 7359: 6969: 6893: 6783:Arithmetic moments 6757: 6707: 6541: 6517: 6462: 6407: 6345: 6261: 6249: 6223: 6184: 6152: 6111: 6073: 6041: 6029:Lambert W function 6017: 5994: 5778: 5758: 5719: 5581: 5530: 5510: 5467: 5465: 5387: 5283: 5113: 5020: 4998: 4976: 4920: 4847: 4686: 4632: 4603: 4508: 4506: 3892: 3841: 3815: 3786: 3720: 3684: 3664: 3638: 3571: 3537: 3496: 3463: 3417: 3369: 3341: 3322: 3266: 3233: 3215: 3194: 3156: 3135: 3115: 3085: 3053: 3017: 2984: 2943: 2898: 2861: 2849: 2810: 2762: 2736:standard deviation 2716: 2696: 2673: 2628: 2602: 2582: 2558: 2379:probability theory 2362: 2240: 2087:Expected shortfall 2072: 1949: 1782: 1773: 1686:Fisher information 1667: 1513: 1498: 1406: 1252: 1145: 1029: 960: 908: 832: 768: 664: 502: 364: 298: 261: 197: 128: 82: 62: 30: 28140: 28139: 27737: 27736: 27706: 27705: 27597:whose type varies 27543:Normal (Gaussian) 27497:Hyperbolic secant 27446:Exponential power 27349:Maxwell–Boltzmann 27097:Wigner semicircle 26989: 26988: 26961:Parabolic fractal 26951:Negative binomial 26705:(12): 4539–4548. 26640:978-0-8247-7803-3 26410:978-0-470-97982-2 26091:Clementi, Fabio; 26067:978-90-70754-33-4 25902:(16): 9439–9449. 25544:(11): 1826–1839. 25501:978-0-486-61114-3 25254:Acta Astronautica 25189:(11): 2143–2150. 25128:10.3390/e19020056 24900:978-1-5386-3428-8 24825:(16): 2709–2714. 24777:Gao, Xin (2009). 24418:978-0-632-04763-5 24156:Journal of Vision 23996:(12): 4539–4548. 23963:978-1-118-12237-2 23937:978-1-4419-5822-8 23783:978-0-471-58495-7 23596: 23593: 23544: 23495: 23463: 23383: 23360: 23338: 23311: 23309: 23299: 23250: 23201: 23169: 23124: 23104: 23074: 23052: 23043: 22906: 22764: 22722: 22672: 22642: 22608: 22454: 22433: 22333: 22315: 22127:Benoit Mandelbrot 22120:, changes in the 22058: 22057: 21985: 21677: 21582: 21581: 21539:In applications, 21510: 21461: 21412: 21380: 21346: 21294: 21199: 21176: 21154: 21120: 21100: 21070: 21004: 21002: 20953: 20904: 20872: 20839: 20797: 20769: 20749: 20727: 20694: 20690: 20648: 20603: 20529: 20527: 20478: 20429: 20397: 20364: 20322: 20294: 20274: 20252: 20205: 20156: 20107: 20075: 20004: 19933: 19905: 19885: 19863: 19741: 19719: 19676: 19674: 19625: 19576: 19544: 19511: 19469: 19441: 19421: 19399: 19366: 19362: 19320: 19275: 19156: 19106: 19102: 19060: 19015: 18930: 18928: 18903: 18877: 18846: 18824: 18565: 18513: 18457: 18455: 18418: 18397: 18373: 18352: 18321: 18250: 18248: 18211: 18190: 18166: 18145: 18113: 18064: 18027: 18007: 17975: 17964: 17943: 17903: 17866: 17840: 17829: 17790: 17738: 17651: 17616: 17587: 17557: 17486: 17484: 17447: 17426: 17402: 17381: 17310: 17284: 17240: 17218: 17188: 17179: 17111: 17084: 16990: 16906: 16893: 16850: 16824: 16763:{\displaystyle =} 16605:{\displaystyle =} 16342: 16317: 16266: 16265: 16256: 16232: 16213: 16074:Method of moments 16052: 15955: 15849: 15831: 15787: 15772: 15756: 15726: 15717: 15394: 15199: 15127:, we can use the 15057: 15031: 14342: 14219: 14116: 14081: 14053: 13985: 13960: 13871: 13659: 13539: 13349:{\displaystyle Z} 13329:{\displaystyle Y} 12875: 12837:{\displaystyle A} 12817:{\displaystyle G} 12797:{\displaystyle H} 12636: 12498:{\displaystyle n} 12413: 12308:{\displaystyle n} 12215:{\displaystyle n} 11901: 11781: 11755: 11636: 11435: 11262: 11024: 10949: 10948: 10850: 10847: 10682: 10598: 10595: 10448: 10424: 10414: 10413: 10338: 10252: 10249: 10239: 10129: 10064: 10061: 9976: 9920: 9917: 9907: 9832: 9769: 9766: 9588: 9581: 9530: 9480: 9416: 9351: 9262: 9255: 9206: 9148: 9085: 9078: 9035: 8977: 8899:{\displaystyle k} 8879:{\displaystyle X} 8799: 8740: 8451:{\displaystyle k} 8431:{\displaystyle X} 8396: 8162:{\displaystyle X} 7929: 7860: 7770: 7769: 7683: 7682: 7565: 7499: 7398: 7358: 7333: 7288: 6968: 6866: 6743: 6702: 6662: 6613: 6551:itself (see also 6076:{\displaystyle y} 6020:{\displaystyle W} 5992: 5991: 5938: 5781:{\displaystyle t} 5673: 5533:{\displaystyle t} 5464: 5209: 5128:covariance matrix 5093: 4840: 4837: 4793: 4771: 4768: 4717: 4685: 4676: 4667: 4651: 4631: 4625: 4597: 4500: 4491: 4479: 4471: 4440: 4419: 4417: 4413: 4412: 4406: 4395: 4389: 4369: 4367: 4361: 4355: 4341: 4336: 4318: 4287: 4282: 4264: 4253: 4228: 4180: 4175: 4157: 4139: 4102: 4078: 4050: 4046: 4009: 3997: 3969: 3965: 3891: 3885: 3873: 3860: 3840: 3834: 3814: 3808: 3716: 3703: 3663: 3657: 3637: 3629: 3610: 3590: 3570: 3564: 3536: 3523: 3495: 3482: 3462: 3436: 3416: 3390: 3365: 3356: 3339: 3321: 3285: 3265: 3257: 3255: 3251: 3250: 3212: 3175: 3152: 3134: 3049: 3036: 3016: 3003: 2980: 2962: 2942: 2917: 2897: 2872: 2848: 2829: 2809: 2788: 2761: 2755: 2557: 2551: 2375: 2374: 2311: 2291: 2261: 2238: 2233: 2187: 2185: 2181: 2180: 2172: 2146: 2137: 2109: 2100: 2070: 2069: 2061: 2051: 2049: 2020: 2015: 2012: 1994: 1969: 1945: 1937: 1935: 1931: 1930: 1920: 1918: 1889: 1884: 1881: 1863: 1855: 1850: 1809: 1797:Method of moments 1781: 1769: 1767: 1754: 1730: 1728: 1715: 1698: 1666: 1624: 1615: 1605: 1589: 1580: 1553: 1512: 1504: 1497: 1476: 1470: 1466: 1465: 1454: 1433: 1405: 1367: 1343: 1326: 1302: 1285: 1279: 1250: 1213: 1180: 1172: 1144: 1120: 1103: 1095: 1064: 1056: 1028: 1020: 1001: 987: 959: 953: 947: 935: 907: 899: 895: 873: 859: 767: 724: 691: 658: 609: 606: 605: 592: 574: 546: 541: 535: 529: 495: 468: 428: 424: 422: 418: 417: 400: 391: 363: 357: 342: 330: 297: 285: 260: 254: 236: 224: 196: 188: 168: 154: 138: 127: 115: 92: 61: 55: 16:(Redirected from 28180: 28130: 28129: 28120: 28119: 28059:Compound Poisson 28034: 28022: 27991:von Mises–Fisher 27987: 27975: 27963: 27925:Circular uniform 27921: 27841: 27785: 27756: 27717: 27716: 27619:Marchenko–Pastur 27482:Geometric stable 27399:Truncated normal 27292:Inverse Gaussian 27198:Hyperexponential 27037:Beta rectangular 27005:bounded interval 27000: 26999: 26868:Discrete uniform 26853:Poisson binomial 26804: 26803: 26779: 26772: 26765: 26756: 26755: 26739: 26714: 26693: 26691: 26659: 26617: 26614: 26608: 26605: 26599: 26598: 26596: 26584: 26578: 26577: 26567: 26543: 26537: 26536: 26510: 26490: 26484: 26483: 26443: 26437: 26436: 26431:. Archived from 26421: 26415: 26414: 26396: 26390: 26389: 26357: 26351: 26350: 26333:(5): 1429–1451. 26318: 26312: 26311: 26301: 26283: 26259: 26253: 26252: 26234: 26214: 26208: 26201: 26195: 26194: 26174: 26168: 26167: 26139: 26133: 26132: 26122: 26120:cond-mat/0202388 26106: 26100: 26093:Gallegati, Mauro 26089: 26083: 26078: 26072: 26071: 26049: 26040: 26034: 26033: 25993: 25987: 25986: 25976: 25936: 25930: 25929: 25919: 25887: 25881: 25880: 25870: 25860: 25836: 25830: 25829: 25819: 25779: 25773: 25772: 25762: 25737:(5): 1010–1021. 25722: 25716: 25715: 25707: 25701: 25700: 25660: 25654: 25653: 25625: 25619: 25618: 25600: 25576: 25570: 25569: 25550:10.1002/sim.2376 25533: 25527: 25520: 25514: 25513: 25487: 25481: 25480: 25478: 25476: 25462: 25456: 25455: 25445: 25439: 25438: 25435:EPJ Data Science 25430: 25419: 25418: 25408: 25368: 25359: 25358: 25348: 25324: 25313: 25312: 25292: 25286: 25285: 25249: 25243: 25242: 25232: 25198: 25174: 25168: 25167: 25139: 25133: 25132: 25130: 25098: 25089: 25084: 25078: 25075: 25069: 25066: 25060: 25054: 25048: 25039: 25028: 25027: 25007: 25001: 25000: 24975:(1/2): 149–176. 24959: 24953: 24952: 24932: 24926: 24925: 24923: 24911: 24905: 24904: 24884: 24868: 24862: 24861: 24852:(9): 2081–2089. 24841: 24835: 24834: 24816: 24807: 24801: 24800: 24798: 24774: 24765: 24764: 24736: 24730: 24729: 24709: 24703: 24702: 24693:(4): 1139–1142. 24682: 24676: 24670: 24664: 24661: 24655: 24654: 24626: 24620: 24619: 24591: 24585: 24584: 24582: 24558: 24552: 24549: 24543: 24540: 24531: 24530: 24520: 24488: 24482: 24481: 24479: 24477: 24467: 24461: 24452: 24446: 24442: 24440: 24432: 24430: 24396: 24390: 24389: 24379: 24347: 24341: 24323: 24317: 24316: 24306: 24296: 24272: 24263: 24262: 24260: 24236: 24230: 24229: 24209: 24200: 24199: 24189: 24171: 24147: 24136: 24127: 24121: 24120: 24118: 24100: 24091: 24085: 24084: 24082: 24056: 24047: 24041: 24040: 24012: 24006: 24005: 23985: 23976: 23975: 23947: 23941: 23940: 23929: 23913: 23907: 23906: 23904: 23897: 23886: 23880: 23879: 23877: 23870: 23859: 23853: 23852: 23850: 23849: 23843: 23837:. Archived from 23828: 23810: 23801: 23795: 23794: 23765: 23750: 23749: 23747: 23746: 23735:www.itl.nist.gov 23727: 23718: 23717: 23715: 23714: 23699: 23686: 23685: 23683: 23682: 23676: 23653: 23633: 23624: 23608: 23607: 23605: 23604: 23599: 23597: 23595: 23594: 23592: 23582: 23581: 23564: 23559: 23550: 23545: 23543: 23533: 23532: 23515: 23510: 23501: 23496: 23494: 23493: 23483: 23478: 23469: 23464: 23462: 23461: 23451: 23446: 23437: 23431: 23429: 23425: 23424: 23419: 23414: 23401: 23396: 23384: 23376: 23368: 23367: 23362: 23361: 23353: 23346: 23345: 23340: 23339: 23331: 23312: 23310: 23305: 23301: 23300: 23298: 23288: 23287: 23270: 23265: 23256: 23251: 23249: 23239: 23238: 23221: 23216: 23207: 23202: 23200: 23199: 23189: 23184: 23175: 23170: 23168: 23167: 23157: 23152: 23143: 23136: 23131: 23129: 23125: 23123: 23119: 23118: 23106: 23105: 23097: 23093: 23089: 23088: 23076: 23075: 23067: 23063: 23054: 23053: 23048: 23044: 23042: 23038: 23037: 23021: 23017: 23016: 23000: 22994: 22983: 22977: 22975: 22973: 22972: 22967: 22965: 22964: 22952: 22951: 22947: 22946: 22914: 22913: 22908: 22907: 22899: 22882: 22880: 22879: 22874: 22872: 22871: 22861: 22860: 22848: 22819: 22805: 22804: 22772: 22771: 22766: 22765: 22757: 22740: 22738: 22737: 22732: 22730: 22729: 22724: 22723: 22715: 22704: 22702: 22701: 22696: 22694: 22693: 22680: 22679: 22674: 22673: 22665: 22658: 22650: 22649: 22644: 22643: 22635: 22616: 22615: 22610: 22609: 22601: 22574: 22573: 22548: 22546: 22545: 22540: 22526: 22525: 22513: 22484: 22483: 22456: 22455: 22447: 22441: 22440: 22435: 22434: 22426: 22403: 22401: 22400: 22395: 22387: 22364: 22363: 22351: 22350: 22335: 22334: 22326: 22323: 22322: 22317: 22316: 22308: 22291: 22285: 22282: 22235:physical testing 22227:Internet traffic 22108: 22106: 22105: 22100: 22079: 22077: 22076: 22071: 22063: 22059: 22053: 22049: 22020: 22018: 22017: 22012: 22000: 21998: 21997: 21992: 21990: 21986: 21978: 21952:Gini coefficient 21949: 21947: 21946: 21941: 21814:surgery duration 21760:maximum rate of 21708:reference ranges 21688: 21686: 21685: 21680: 21678: 21673: 21671: 21670: 21645: 21643: 21642: 21637: 21625: 21623: 21622: 21617: 21593: 21591: 21590: 21585: 21583: 21577: 21573: 21558: 21556: 21555: 21550: 21528: 21526: 21525: 21520: 21518: 21517: 21516: 21512: 21511: 21509: 21499: 21498: 21481: 21476: 21467: 21462: 21460: 21450: 21449: 21432: 21427: 21418: 21413: 21411: 21410: 21400: 21395: 21386: 21381: 21379: 21378: 21368: 21363: 21354: 21347: 21339: 21330: 21325: 21312: 21307: 21295: 21287: 21279: 21278: 21266: 21265: 21245: 21241: 21240: 21235: 21230: 21217: 21212: 21200: 21192: 21184: 21183: 21178: 21177: 21169: 21162: 21161: 21156: 21155: 21147: 21125: 21121: 21119: 21115: 21114: 21102: 21101: 21093: 21089: 21085: 21084: 21072: 21071: 21063: 21059: 21022: 21020: 21019: 21014: 21012: 21011: 21010: 21006: 21005: 21003: 21001: 20991: 20990: 20973: 20968: 20959: 20954: 20952: 20942: 20941: 20924: 20919: 20910: 20905: 20903: 20902: 20892: 20887: 20878: 20873: 20871: 20870: 20860: 20855: 20846: 20844: 20842: 20841: 20840: 20832: 20812: 20807: 20798: 20790: 20784: 20779: 20770: 20762: 20757: 20756: 20751: 20750: 20742: 20735: 20734: 20729: 20728: 20720: 20700: 20696: 20695: 20693: 20692: 20691: 20685: 20680: 20671: 20666: 20665: 20651: 20650: 20649: 20643: 20638: 20629: 20624: 20623: 20609: 20604: 20602: 20598: 20597: 20581: 20577: 20576: 20560: 20540: 20538: 20537: 20532: 20530: 20528: 20526: 20516: 20515: 20498: 20493: 20484: 20479: 20477: 20467: 20466: 20449: 20444: 20435: 20430: 20428: 20427: 20417: 20412: 20403: 20398: 20396: 20395: 20385: 20380: 20371: 20369: 20367: 20366: 20365: 20357: 20337: 20332: 20323: 20315: 20309: 20304: 20295: 20287: 20282: 20281: 20276: 20275: 20267: 20260: 20259: 20254: 20253: 20245: 20230:Pivotal quantity 20221: 20219: 20218: 20213: 20211: 20207: 20206: 20204: 20194: 20193: 20176: 20171: 20162: 20157: 20155: 20145: 20144: 20127: 20122: 20113: 20108: 20106: 20105: 20095: 20090: 20081: 20076: 20074: 20073: 20063: 20058: 20049: 20040: 20035: 20022: 20017: 20005: 19997: 19989: 19988: 19976: 19975: 19948: 19943: 19934: 19926: 19920: 19915: 19906: 19898: 19893: 19892: 19887: 19886: 19878: 19871: 19870: 19865: 19864: 19856: 19843:This means that 19823: 19821: 19820: 19815: 19812: 19807: 19791: 19789: 19788: 19783: 19780: 19775: 19759: 19757: 19756: 19751: 19749: 19748: 19743: 19742: 19734: 19727: 19726: 19721: 19720: 19712: 19694: 19692: 19691: 19686: 19684: 19683: 19682: 19678: 19677: 19675: 19673: 19663: 19662: 19645: 19640: 19631: 19626: 19624: 19614: 19613: 19596: 19591: 19582: 19577: 19575: 19574: 19564: 19559: 19550: 19545: 19543: 19542: 19532: 19527: 19518: 19516: 19514: 19513: 19512: 19504: 19484: 19479: 19470: 19462: 19456: 19451: 19442: 19434: 19429: 19428: 19423: 19422: 19414: 19407: 19406: 19401: 19400: 19392: 19372: 19368: 19367: 19365: 19364: 19363: 19357: 19352: 19343: 19338: 19337: 19323: 19322: 19321: 19315: 19310: 19301: 19296: 19295: 19281: 19276: 19274: 19270: 19269: 19253: 19249: 19248: 19232: 19208: 19206: 19205: 19200: 19198: 19197: 19192: 19187: 19174: 19169: 19157: 19149: 19141: 19140: 19128: 19127: 19107: 19105: 19104: 19103: 19097: 19092: 19083: 19078: 19077: 19063: 19062: 19061: 19055: 19050: 19041: 19036: 19035: 19021: 19016: 19014: 19010: 19009: 18993: 18989: 18988: 18972: 18948: 18946: 18945: 18940: 18938: 18937: 18936: 18932: 18931: 18929: 18923: 18918: 18909: 18904: 18898: 18893: 18884: 18882: 18880: 18879: 18878: 18870: 18854: 18853: 18848: 18847: 18839: 18832: 18831: 18826: 18825: 18817: 18797: 18796: 18795: 18794: 18782: 18781: 18746: 18744: 18743: 18738: 18736: 18735: 18719: 18717: 18716: 18711: 18709: 18708: 18692: 18690: 18689: 18684: 18678: 18673: 18661: 18660: 18641: 18639: 18638: 18633: 18627: 18622: 18610: 18609: 18578: 18576: 18575: 18570: 18568: 18567: 18566: 18558: 18526: 18524: 18523: 18518: 18516: 18515: 18514: 18506: 18475: 18473: 18472: 18467: 18465: 18464: 18463: 18459: 18458: 18456: 18454: 18434: 18433: 18424: 18419: 18414: 18413: 18404: 18402: 18400: 18399: 18398: 18390: 18374: 18369: 18368: 18359: 18354: 18353: 18345: 18329: 18325: 18324: 18323: 18322: 18317: 18316: 18307: 18261: 18259: 18258: 18253: 18251: 18249: 18247: 18227: 18226: 18217: 18212: 18207: 18206: 18197: 18195: 18193: 18192: 18191: 18183: 18167: 18162: 18161: 18152: 18147: 18146: 18138: 18127:Pivotal quantity 18124: 18122: 18121: 18116: 18114: 18109: 18108: 18099: 18080: 18078: 18077: 18072: 18070: 18066: 18065: 18063: 18043: 18042: 18033: 18028: 18023: 18022: 18013: 18008: 18003: 18002: 17993: 17977: 17976: 17968: 17965: 17960: 17959: 17950: 17945: 17944: 17936: 17919: 17917: 17916: 17911: 17909: 17905: 17904: 17902: 17882: 17881: 17872: 17867: 17862: 17861: 17852: 17842: 17841: 17833: 17830: 17825: 17824: 17815: 17806: 17804: 17803: 17798: 17796: 17792: 17791: 17789: 17778: 17777: 17776: 17763: 17758: 17757: 17740: 17739: 17731: 17728: 17727: 17695: 17693: 17692: 17687: 17685: 17684: 17667: 17665: 17664: 17659: 17657: 17653: 17652: 17647: 17646: 17637: 17618: 17617: 17609: 17599: 17597: 17596: 17591: 17589: 17588: 17580: 17570: 17568: 17567: 17562: 17560: 17559: 17558: 17553: 17552: 17543: 17504: 17502: 17501: 17496: 17494: 17493: 17492: 17488: 17487: 17485: 17483: 17463: 17462: 17453: 17448: 17443: 17442: 17433: 17431: 17429: 17428: 17427: 17419: 17403: 17398: 17397: 17388: 17383: 17382: 17374: 17321: 17319: 17318: 17313: 17311: 17309: 17298: 17297: 17296: 17291: 17287: 17286: 17285: 17277: 17271: 17270: 17248: 17238: 17233: 17232: 17219: 17214: 17213: 17212: 17196: 17186: 17181: 17180: 17172: 17155: 17153: 17152: 17147: 17124: 17122: 17121: 17116: 17114: 17113: 17112: 17107: 17105: 17092: 17091: 17086: 17085: 17077: 17067: 17066: 17050: 17048: 17047: 17044:{\displaystyle } 17042: 17037: 17036: 17027: 17026: 17014: 17007: 17006: 17001: 16998: 16997: 16992: 16991: 16983: 16969: 16967: 16966: 16961: 16959: 16958: 16946: 16945: 16917: 16915: 16914: 16909: 16907: 16902: 16900: 16895: 16894: 16886: 16880: 16865: 16863: 16862: 16859:{\displaystyle } 16857: 16852: 16851: 16846: 16838: 16826: 16825: 16817: 16804: 16802: 16801: 16796: 16769: 16767: 16766: 16761: 16756: 16755: 16746: 16745: 16733: 16726: 16725: 16720: 16717: 16716: 16698: 16697: 16688: 16687: 16672: 16671: 16659: 16658: 16649: 16648: 16636: 16631: 16630: 16611: 16609: 16608: 16603: 16598: 16597: 16588: 16581: 16580: 16575: 16572: 16571: 16553: 16552: 16540: 16539: 16527: 16526: 16517: 16512: 16511: 16492: 16490: 16489: 16486:{\displaystyle } 16484: 16442: 16440: 16439: 16436:{\displaystyle } 16434: 16368: 16366: 16365: 16360: 16355: 16351: 16350: 16349: 16344: 16343: 16335: 16331: 16326: 16325: 16324: 16319: 16318: 16310: 16285: 16284: 16271: 16267: 16264: 16263: 16258: 16257: 16249: 16245: 16240: 16239: 16234: 16233: 16225: 16215: 16214: 16206: 16204: 16179: 16177: 16176: 16171: 16159: 16157: 16156: 16151: 16139: 16137: 16136: 16131: 16107: 16105: 16104: 16099: 16064: 16062: 16061: 16056: 16054: 16053: 16045: 16035: 16033: 16032: 16027: 16025: 16024: 16006: 16005: 15993: 15992: 15973: 15971: 15970: 15965: 15963: 15962: 15957: 15956: 15948: 15929: 15927: 15926: 15921: 15909: 15907: 15906: 15901: 15889: 15887: 15886: 15881: 15863: 15861: 15860: 15855: 15850: 15845: 15844: 15843: 15838: 15834: 15833: 15832: 15824: 15818: 15817: 15795: 15785: 15780: 15779: 15774: 15773: 15765: 15757: 15752: 15751: 15750: 15734: 15724: 15719: 15718: 15710: 15700: 15698: 15697: 15692: 15687: 15686: 15662: 15661: 15643: 15642: 15620: 15618: 15617: 15612: 15600: 15598: 15597: 15592: 15580: 15578: 15577: 15572: 15570: 15569: 15553: 15551: 15550: 15545: 15523: 15521: 15520: 15515: 15507: 15506: 15482: 15481: 15463: 15462: 15432: 15431: 15419: 15418: 15402: 15384: 15383: 15365: 15364: 15352: 15351: 15317: 15315: 15314: 15309: 15304: 15303: 15285: 15284: 15271: 15269: 15268: 15263: 15251: 15249: 15248: 15243: 15235: 15234: 15216: 15215: 15200: 15198: 15197: 15185: 15182: 15177: 15096: 15094: 15093: 15088: 15083: 15082: 15074: 15070: 15063: 15062: 15058: 15053: 15045: 15036: 15032: 15027: 15026: 15017: 14956: 14954: 14953: 14948: 14912: 14910: 14909: 14904: 14899: 14898: 14861: 14859: 14858: 14853: 14807: 14805: 14804: 14799: 14751: 14749: 14748: 14743: 14689: 14687: 14686: 14681: 14644: 14642: 14641: 14636: 14618: 14616: 14615: 14610: 14605: 14604: 14564: 14562: 14561: 14556: 14554: 14547: 14541: 14536: 14524: 14520: 14519: 14493: 14492: 14479: 14478: 14474: 14468: 14463: 14451: 14450: 14436: 14435: 14431: 14425: 14420: 14408: 14407: 14393: 14392: 14383: 14382: 14370: 14369: 14356: 14339: 14334: 14325: 14311: 14310: 14289: 14288: 14270: 14269: 14260: 14259: 14247: 14246: 14233: 14216: 14211: 14202: 14186: 14181: 14168: 14167: 14163: 14157: 14152: 14140: 14139: 14124: 14109: 14108: 14089: 14077: 14073: 14072: 14071: 14061: 14034: 14033: 14003: 14001: 14000: 13995: 13993: 13986: 13980: 13975: 13966: 13961: 13956: 13955: 13946: 13941: 13937: 13936: 13935: 13934: 13933: 13900: 13899: 13883: 13879: 13872: 13870: 13869: 13868: 13859: 13858: 13857: 13856: 13835: 13834: 13833: 13832: 13831: 13810: 13799: 13798: 13797: 13796: 13762: 13757: 13737: 13735: 13734: 13729: 13721: 13720: 13704: 13702: 13701: 13696: 13694: 13693: 13677: 13675: 13674: 13669: 13667: 13660: 13654: 13649: 13640: 13635: 13631: 13630: 13629: 13625: 13619: 13614: 13602: 13601: 13568: 13567: 13551: 13547: 13540: 13538: 13537: 13536: 13527: 13526: 13522: 13516: 13511: 13499: 13498: 13477: 13467: 13466: 13464: 13459: 13442: 13441: 13439: 13434: 13422: 13421: 13400: 13381: 13376: 13355: 13353: 13352: 13347: 13335: 13333: 13332: 13327: 13315: 13313: 13312: 13307: 13305: 13304: 13294: 13289: 13262:parameters, and 13261: 13259: 13258: 13253: 13241: 13239: 13238: 13233: 13221: 13219: 13218: 13213: 13207: 13202: 13190: 13189: 13168: 13167: 13148: 13146: 13145: 13140: 13135: 13134: 13116: 13115: 13084: 13082: 13081: 13076: 13071: 13070: 13031: 13029: 13028: 13023: 13015: 13014: 12961: 12959: 12958: 12953: 12948: 12947: 12929: 12928: 12889: 12887: 12886: 12881: 12876: 12871: 12870: 12861: 12843: 12841: 12840: 12835: 12823: 12821: 12820: 12815: 12803: 12801: 12800: 12795: 12753: 12751: 12750: 12745: 12740: 12739: 12711: 12709: 12708: 12703: 12701: 12700: 12684: 12682: 12681: 12676: 12671: 12660: 12659: 12638: 12634: 12628: 12627: 12611: 12609: 12608: 12603: 12595: 12594: 12557: 12555: 12554: 12549: 12531: 12529: 12528: 12523: 12521: 12520: 12504: 12502: 12501: 12496: 12471: 12469: 12468: 12463: 12456: 12455: 12448: 12443: 12433: 12428: 12411: 12407: 12406: 12396: 12391: 12375: 12374: 12359: 12358: 12348: 12343: 12314: 12312: 12311: 12306: 12294: 12292: 12291: 12286: 12280: 12275: 12263: 12262: 12241: 12240: 12222:such variables. 12221: 12219: 12218: 12213: 12201: 12199: 12198: 12193: 12190: 12185: 12172: 12167: 12155: 12154: 12138: 12136: 12135: 12130: 12118: 12116: 12115: 12110: 12108: 12107: 12095: 12094: 12072: 12070: 12069: 12064: 12062: 12061: 12049: 12048: 12026: 12024: 12023: 12018: 12016: 12015: 11999: 11997: 11996: 11991: 11989: 11988: 11960: 11958: 11957: 11952: 11934: 11932: 11931: 11926: 11921: 11920: 11911: 11910: 11899: 11877: 11876: 11860: 11858: 11857: 11852: 11847: 11846: 11807: 11805: 11804: 11799: 11791: 11790: 11779: 11757: 11748: 11738: 11736: 11735: 11730: 11725: 11724: 11685: 11683: 11682: 11677: 11659: 11657: 11656: 11651: 11646: 11645: 11634: 11590: 11588: 11587: 11582: 11577: 11576: 11527: 11525: 11524: 11519: 11513: 11508: 11499: 11493: 11488: 11451: 11449: 11448: 11443: 11441: 11437: 11436: 11433: 11428: 11419: 11413: 11408: 11393: 11391: 11386: 11385: 11352: 11350: 11349: 11344: 11315: 11314: 11302: 11301: 11273: 11271: 11270: 11265: 11263: 11240: 11232: 11206: 11205: 11189: 11187: 11186: 11181: 11173: 11147: 11146: 11130: 11128: 11127: 11122: 11117: 11116: 11104: 11103: 11040: 11038: 11037: 11032: 11030: 11026: 11025: 11023: 11018: 11013: 11004: 10998: 10993: 10965: 10964: 10963: 10958: 10957: 10950: 10946: 10941: 10932: 10926: 10921: 10906: 10905: 10904: 10895: 10875: 10874: 10867: 10851: 10849: 10848: 10846: 10842: 10840: 10835: 10826: 10820: 10815: 10783: 10774: 10766: 10765: 10764: 10751: 10750: 10749: 10698: 10696: 10695: 10690: 10688: 10684: 10683: 10681: 10677: 10676: 10661: 10660: 10647: 10640: 10626: 10625: 10615: 10599: 10597: 10596: 10588: 10583: 10582: 10557: 10549: 10548: 10547: 10534: 10492: 10490: 10489: 10484: 10482: 10478: 10477: 10476: 10449: 10441: 10425: 10417: 10415: 10412: 10401: 10400: 10392: 10384: 10354: 10352: 10351: 10346: 10344: 10340: 10339: 10337: 10327: 10326: 10301: 10294: 10280: 10279: 10269: 10253: 10251: 10250: 10242: 10240: 10229: 10228: 10207: 10198: 10190: 10179: 10145: 10143: 10142: 10137: 10135: 10131: 10130: 10128: 10127: 10126: 10113: 10106: 10092: 10091: 10081: 10065: 10063: 10062: 10054: 10042: 10034: 10026: 9992: 9990: 9989: 9984: 9982: 9978: 9977: 9975: 9967: 9966: 9965: 9937: 9921: 9919: 9918: 9910: 9908: 9903: 9894: 9886: 9878: 9848: 9846: 9845: 9840: 9838: 9834: 9833: 9831: 9830: 9829: 9816: 9815: 9814: 9786: 9770: 9768: 9767: 9759: 9747: 9739: 9731: 9677: 9675: 9674: 9669: 9667: 9666: 9654: 9653: 9637: 9635: 9634: 9629: 9603: 9601: 9600: 9595: 9593: 9589: 9587: 9586: 9582: 9577: 9567: 9566: 9547: 9535: 9531: 9526: 9516: 9515: 9496: 9486: 9485: 9481: 9476: 9475: 9474: 9453: 9452: 9433: 9421: 9417: 9412: 9411: 9410: 9389: 9388: 9369: 9359: 9354: 9353: 9352: 9347: 9346: 9337: 9311: 9310: 9298: 9297: 9263: 9261: 9260: 9256: 9251: 9228: 9212: 9211: 9207: 9202: 9183: 9182: 9166: 9156: 9151: 9150: 9149: 9144: 9143: 9134: 9086: 9084: 9083: 9079: 9074: 9051: 9041: 9040: 9036: 9031: 9030: 9029: 8995: 8985: 8980: 8979: 8978: 8973: 8972: 8963: 8905: 8903: 8902: 8897: 8885: 8883: 8882: 8877: 8837: 8835: 8834: 8829: 8814: 8812: 8811: 8806: 8804: 8800: 8795: 8782: 8781: 8765: 8754: 8753: 8752: 8751: 8742: 8733: 8688: 8687: 8674: 8669: 8635: 8633: 8632: 8627: 8551: 8549: 8548: 8543: 8510: 8509: 8496: 8491: 8457: 8455: 8454: 8449: 8437: 8435: 8434: 8429: 8409: 8407: 8406: 8401: 8394: 8393: 8392: 8380: 8379: 8339: 8337: 8336: 8331: 8320: 8319: 8300: 8298: 8297: 8292: 8287: 8286: 8276: 8275: 8261: 8260: 8248: 8247: 8235: 8234: 8224: 8223: 8188: 8187: 8168: 8166: 8165: 8160: 8148: 8146: 8145: 8140: 8109: 8107: 8106: 8101: 8096: 8095: 8094: 8093: 8045: 8043: 8042: 8037: 8029: 7966: 7959: 7952: 7950: 7949: 7944: 7942: 7935: 7931: 7930: 7928: 7927: 7926: 7904: 7887: 7865: 7861: 7859: 7858: 7857: 7835: 7831: 7830: 7811: 7792: 7791: 7775: 7771: 7768: 7767: 7728: 7727: 7726: 7725: 7703: 7688: 7684: 7678: 7677: 7659: 7658: 7657: 7656: 7634: 7598: 7592: 7579: 7577: 7576: 7571: 7566: 7558: 7557: 7556: 7555: 7541: 7511: 7509: 7508: 7503: 7501: 7498: 7481: 7464: 7454: 7452: 7451: 7446: 7416: 7414: 7413: 7408: 7406: 7399: 7391: 7390: 7389: 7388: 7374: 7372: 7371: 7370: 7369: 7360: 7351: 7334: 7326: 7325: 7324: 7323: 7309: 7289: 7272: 7232: 7231: 7230: 7229: 7212: 7211: 7210: 7209: 7174: 7173: 7172: 7171: 7154: 7153: 7123: 7122: 7095: 7094: 7047: 7046: 7045: 7044: 7008: 7007: 6982: 6981: 6980: 6979: 6970: 6961: 6911: 6902: 6900: 6899: 6894: 6889: 6888: 6887: 6886: 6877: 6876: 6867: 6859: 6837: 6836: 6808: 6798: 6792: 6766: 6764: 6763: 6758: 6756: 6755: 6754: 6753: 6744: 6736: 6716: 6714: 6713: 6708: 6703: 6686: 6663: 6661: 6660: 6659: 6658: 6644: 6639: 6638: 6626: 6625: 6624: 6623: 6614: 6606: 6564:concave function 6560:AM–GM inequality 6550: 6548: 6547: 6542: 6526: 6524: 6523: 6518: 6510: 6509: 6471: 6469: 6468: 6463: 6461: 6460: 6459: 6458: 6416: 6414: 6413: 6408: 6406: 6405: 6393: 6392: 6354: 6352: 6351: 6346: 6344: 6343: 6331: 6330: 6258: 6256: 6255: 6250: 6232: 6230: 6229: 6224: 6216: 6215: 6193: 6191: 6190: 6185: 6161: 6159: 6158: 6153: 6120: 6118: 6117: 6112: 6082: 6080: 6079: 6074: 6050: 6048: 6047: 6042: 6026: 6024: 6023: 6018: 6003: 6001: 6000: 5995: 5993: 5987: 5986: 5977: 5976: 5946: 5945: 5944: 5940: 5939: 5937: 5936: 5935: 5922: 5918: 5917: 5908: 5907: 5874: 5873: 5864: 5863: 5842: 5841: 5831: 5814: 5787: 5785: 5784: 5779: 5767: 5765: 5764: 5759: 5734:divergent series 5728: 5726: 5725: 5720: 5718: 5717: 5713: 5708: 5707: 5698: 5697: 5674: 5672: 5664: 5663: 5662: 5643: 5640: 5635: 5602: 5596: 5590: 5588: 5587: 5582: 5577: 5576: 5539: 5537: 5536: 5531: 5519: 5517: 5516: 5511: 5506: 5505: 5476: 5474: 5473: 5468: 5466: 5460: 5456: 5455: 5415: 5396: 5394: 5393: 5388: 5386: 5385: 5381: 5376: 5375: 5366: 5365: 5336: 5335: 5292: 5290: 5289: 5284: 5270: 5269: 5268: 5267: 5247: 5246: 5242: 5241: 5226: 5225: 5210: 5202: 5197: 5196: 5184: 5183: 5166: 5165: 5153: 5122: 5120: 5119: 5114: 5109: 5108: 5107: 5106: 5094: 5086: 5081: 5080: 5063: 5062: 5053: 5029: 5027: 5026: 5021: 5019: 5007: 5005: 5004: 4999: 4997: 4985: 4983: 4982: 4977: 4972: 4971: 4950: 4949: 4929: 4927: 4926: 4921: 4916: 4907: 4899: 4898: 4889: 4865: 4864: 4856: 4854: 4853: 4848: 4846: 4842: 4841: 4839: 4838: 4833: 4827: 4810: 4794: 4786: 4781: 4777: 4776: 4772: 4770: 4769: 4764: 4758: 4741: 4718: 4710: 4695: 4693: 4692: 4687: 4683: 4674: 4665: 4658: 4657: 4649: 4641: 4639: 4638: 4633: 4629: 4623: 4612: 4610: 4609: 4604: 4602: 4598: 4593: 4570: 4549: 4548: 4517: 4515: 4514: 4509: 4507: 4498: 4497: 4493: 4492: 4490: 4489: 4488: 4477: 4472: 4469: 4468: 4467: 4438: 4436: 4420: 4418: 4415: 4414: 4410: 4404: 4400: 4393: 4387: 4382: 4374: 4370: 4368: 4365: 4359: 4353: 4348: 4346: 4342: 4337: 4334: 4316: 4314: 4307: 4296: 4292: 4288: 4283: 4280: 4262: 4260: 4254: 4252: 4248: 4247: 4240: 4235: 4233: 4229: 4224: 4207: 4200: 4189: 4185: 4181: 4176: 4173: 4155: 4153: 4145: 4140: 4138: 4134: 4133: 4126: 4121: 4113: 4109: 4108: 4100: 4076: 4075: 4074: 4066: 4065: 4064: 4063: 4057: 4048: 4047: 4045: 4041: 4040: 4033: 4028: 4020: 4016: 4015: 4007: 3995: 3994: 3993: 3985: 3984: 3983: 3982: 3976: 3967: 3966: 3964: 3960: 3959: 3952: 3947: 3929: 3928: 3901: 3899: 3898: 3893: 3889: 3883: 3871: 3867: 3866: 3858: 3850: 3848: 3847: 3842: 3838: 3832: 3824: 3822: 3821: 3816: 3812: 3806: 3795: 3793: 3792: 3787: 3782: 3781: 3763: 3762: 3729: 3727: 3726: 3721: 3714: 3713: 3712: 3701: 3693: 3691: 3690: 3685: 3673: 3671: 3670: 3665: 3661: 3655: 3647: 3645: 3644: 3639: 3635: 3634: 3630: 3627: 3626: 3625: 3608: 3588: 3580: 3578: 3577: 3572: 3568: 3562: 3546: 3544: 3543: 3538: 3534: 3533: 3532: 3521: 3505: 3503: 3502: 3497: 3493: 3492: 3491: 3480: 3472: 3470: 3469: 3464: 3460: 3459: 3458: 3446: 3445: 3434: 3426: 3424: 3423: 3418: 3414: 3413: 3412: 3400: 3399: 3388: 3378: 3376: 3375: 3370: 3363: 3362: 3358: 3357: 3354: 3349: 3340: 3337: 3335: 3330: 3319: 3317: 3295: 3294: 3283: 3275: 3273: 3272: 3267: 3263: 3262: 3258: 3256: 3253: 3252: 3248: 3246: 3241: 3228: 3223: 3214: 3210: 3207: 3202: 3193: 3173: 3165: 3163: 3162: 3157: 3150: 3148: 3143: 3132: 3124: 3122: 3121: 3116: 3114: 3113: 3094: 3092: 3091: 3086: 3062: 3060: 3059: 3054: 3047: 3046: 3045: 3034: 3026: 3024: 3023: 3018: 3014: 3013: 3012: 3001: 2993: 2991: 2990: 2985: 2978: 2960: 2952: 2950: 2949: 2944: 2940: 2927: 2926: 2915: 2907: 2905: 2904: 2899: 2895: 2882: 2881: 2870: 2858: 2856: 2855: 2850: 2846: 2845: 2844: 2827: 2819: 2817: 2816: 2811: 2807: 2786: 2771: 2769: 2768: 2763: 2759: 2753: 2726:. These are the 2725: 2723: 2722: 2717: 2705: 2703: 2702: 2697: 2682: 2680: 2679: 2674: 2672: 2671: 2637: 2635: 2634: 2629: 2611: 2609: 2608: 2603: 2591: 2589: 2588: 2583: 2567: 2565: 2564: 2559: 2555: 2549: 2528: 2520: 2503:random variables 2451: 2449: 2437: 2429: 2425: 2424: 2412: 2393:is a continuous 2371: 2369: 2368: 2363: 2340: 2339: 2312: 2310: 2296: 2294: 2293: 2292: 2287: 2286: 2277: 2259: 2249: 2247: 2246: 2241: 2239: 2234: 2231: 2230: 2226: 2204: 2203: 2188: 2186: 2183: 2182: 2178: 2174: 2170: 2165: 2144: 2142: 2140: 2139: 2138: 2133: 2132: 2123: 2110: 2102: 2098: 2081: 2079: 2078: 2073: 2071: 2067: 2066: 2062: 2059: 2052: 2050: 2047: 2046: 2045: 2027: 2026: 2018: 2016: 2013: 2010: 1992: 1990: 1977: 1967: 1958: 1956: 1955: 1950: 1943: 1942: 1938: 1936: 1933: 1932: 1928: 1921: 1919: 1916: 1915: 1914: 1896: 1895: 1887: 1885: 1882: 1879: 1861: 1859: 1857: 1853: 1851: 1848: 1835: 1834: 1827: 1807: 1791: 1789: 1788: 1783: 1779: 1778: 1777: 1770: 1768: 1765: 1764: 1763: 1752: 1747: 1731: 1729: 1726: 1725: 1724: 1713: 1708: 1696: 1676: 1674: 1673: 1668: 1664: 1663: 1662: 1658: 1653: 1652: 1643: 1642: 1622: 1616: 1614: 1606: 1603: 1602: 1601: 1587: 1578: 1576: 1573: 1568: 1551: 1522: 1520: 1519: 1514: 1510: 1509: 1505: 1502: 1501: 1500: 1499: 1490: 1474: 1468: 1467: 1463: 1456: 1452: 1443: 1442: 1431: 1415: 1413: 1412: 1407: 1403: 1396: 1392: 1391: 1390: 1365: 1358: 1354: 1353: 1352: 1341: 1324: 1317: 1313: 1312: 1311: 1300: 1283: 1277: 1261: 1259: 1258: 1253: 1251: 1239: 1238: 1220: 1218: 1214: 1211: 1204: 1200: 1199: 1178: 1170: 1154: 1152: 1151: 1146: 1142: 1141: 1137: 1136: 1135: 1118: 1101: 1100: 1096: 1093: 1083: 1082: 1062: 1054: 1038: 1036: 1035: 1030: 1026: 1025: 1021: 1018: 1017: 1016: 999: 985: 969: 967: 966: 961: 957: 951: 945: 933: 917: 915: 914: 909: 905: 904: 900: 897: 896: 891: 890: 881: 871: 857: 841: 839: 838: 833: 819: 818: 777: 775: 774: 769: 765: 764: 760: 738: 737: 725: 723: 722: 710: 689: 673: 671: 670: 665: 663: 659: 654: 631: 619: 615: 614: 610: 608: 607: 603: 599: 593: 590: 572: 570: 547: 542: 539: 533: 531: 527: 511: 509: 508: 503: 501: 497: 496: 494: 493: 492: 479: 478: 473: 469: 466: 444: 426: 425: 423: 420: 419: 415: 408: 398: 393: 389: 373: 371: 370: 365: 361: 355: 340: 328: 307: 305: 304: 299: 295: 283: 270: 268: 267: 262: 258: 252: 234: 222: 206: 204: 203: 198: 194: 193: 189: 186: 185: 184: 166: 152: 137: 135: 134: 129: 125: 113: 106: 104: 91: 89: 88: 83: 71: 69: 68: 63: 59: 53: 45: 43: 33: 29: 21: 28188: 28187: 28183: 28182: 28181: 28179: 28178: 28177: 28143: 28142: 28141: 28136: 28108: 28084:Maximum entropy 28042: 28030: 28018: 28008: 28000: 27983: 27971: 27959: 27914: 27901: 27838:Matrix-valued: 27835: 27781: 27752: 27744: 27733: 27721: 27712: 27702: 27596: 27590: 27507: 27433: 27431: 27425: 27354:Maxwell–Jüttner 27203:Hypoexponential 27109: 27107: 27106:supported on a 27101: 27062:Noncentral beta 27022:Balding–Nichols 27004: 27003:supported on a 26995: 26985: 26888: 26882: 26878:Zipf–Mandelbrot 26808: 26799: 26793: 26783: 26746: 26641: 26625: 26623:Further reading 26620: 26615: 26611: 26606: 26602: 26585: 26581: 26544: 26540: 26491: 26487: 26444: 26440: 26423: 26422: 26418: 26411: 26397: 26393: 26358: 26354: 26319: 26315: 26260: 26256: 26215: 26211: 26202: 26198: 26191: 26183:. Basic Books. 26175: 26171: 26140: 26136: 26107: 26103: 26090: 26086: 26079: 26075: 26068: 26047: 26041: 26037: 25994: 25990: 25937: 25933: 25896:Cerebral Cortex 25888: 25884: 25837: 25833: 25780: 25776: 25723: 25719: 25708: 25704: 25661: 25657: 25626: 25622: 25577: 25573: 25534: 25530: 25521: 25517: 25502: 25488: 25484: 25474: 25472: 25464: 25463: 25459: 25446: 25442: 25431: 25422: 25389:10.1038/nrn3687 25369: 25362: 25325: 25316: 25293: 25289: 25250: 25246: 25175: 25171: 25140: 25136: 25099: 25092: 25085: 25081: 25076: 25072: 25067: 25063: 25055: 25051: 25040: 25031: 25008: 25004: 24981:10.2307/2332539 24960: 24956: 24933: 24929: 24912: 24908: 24901: 24869: 24865: 24842: 24838: 24814: 24808: 24804: 24775: 24768: 24737: 24733: 24710: 24706: 24683: 24679: 24671: 24667: 24662: 24658: 24627: 24623: 24592: 24588: 24559: 24555: 24550: 24546: 24541: 24534: 24503:(17): 2719–21. 24489: 24485: 24475: 24473: 24469: 24468: 24464: 24444: 24443: 24434: 24433: 24419: 24397: 24393: 24348: 24344: 24334:Wayback Machine 24324: 24320: 24273: 24266: 24237: 24233: 24210: 24203: 24148: 24139: 24128: 24124: 24098: 24092: 24088: 24054: 24048: 24044: 24013: 24009: 23986: 23979: 23964: 23948: 23944: 23938: 23914: 23910: 23902: 23895: 23887: 23883: 23875: 23868: 23860: 23856: 23847: 23845: 23841: 23826:10.1.1.511.9750 23808: 23802: 23798: 23784: 23766: 23753: 23744: 23742: 23729: 23728: 23721: 23712: 23710: 23700: 23689: 23680: 23678: 23674: 23631: 23625: 23621: 23617: 23612: 23611: 23577: 23573: 23566: 23560: 23555: 23549: 23528: 23524: 23517: 23511: 23506: 23500: 23489: 23485: 23479: 23474: 23468: 23457: 23453: 23447: 23442: 23436: 23435: 23430: 23415: 23410: 23397: 23392: 23375: 23363: 23352: 23351: 23350: 23341: 23330: 23329: 23328: 23324: 23320: 23316: 23283: 23279: 23272: 23266: 23261: 23255: 23234: 23230: 23223: 23217: 23212: 23206: 23195: 23191: 23185: 23180: 23174: 23163: 23159: 23153: 23148: 23142: 23141: 23137: 23135: 23130: 23114: 23110: 23096: 23095: 23094: 23084: 23080: 23066: 23065: 23064: 23062: 23058: 23033: 23029: 23022: 23012: 23008: 23001: 22999: 22995: 22993: 22992: 22990: 22987: 22986: 22984: 22980: 22960: 22956: 22942: 22938: 22925: 22921: 22909: 22898: 22897: 22896: 22888: 22885: 22884: 22856: 22852: 22844: 22815: 22800: 22796: 22783: 22779: 22767: 22756: 22755: 22754: 22746: 22743: 22742: 22725: 22714: 22713: 22712: 22710: 22707: 22706: 22675: 22664: 22663: 22662: 22654: 22645: 22634: 22633: 22632: 22611: 22600: 22599: 22598: 22585: 22581: 22569: 22565: 22554: 22551: 22550: 22521: 22517: 22509: 22479: 22475: 22446: 22445: 22436: 22425: 22424: 22423: 22409: 22406: 22405: 22383: 22359: 22355: 22346: 22342: 22325: 22324: 22318: 22307: 22306: 22305: 22303: 22300: 22299: 22292: 22288: 22283: 22279: 22274: 22268: 22244: 22176: 22085: 22082: 22081: 22048: 22044: 22030: 22027: 22026: 22006: 22003: 22002: 21977: 21973: 21959: 21956: 21955: 21935: 21932: 21931: 21912: 21880:confidence belt 21860: 21826: 21792: 21785:pharmacokinetic 21747: 21724: 21695: 21672: 21666: 21665: 21654: 21651: 21650: 21631: 21628: 21627: 21602: 21599: 21598: 21572: 21564: 21561: 21560: 21544: 21541: 21540: 21537: 21494: 21490: 21483: 21477: 21472: 21466: 21445: 21441: 21434: 21428: 21423: 21417: 21406: 21402: 21396: 21391: 21385: 21374: 21370: 21364: 21359: 21353: 21352: 21348: 21338: 21326: 21321: 21308: 21303: 21286: 21274: 21270: 21261: 21257: 21253: 21249: 21231: 21226: 21213: 21208: 21191: 21179: 21168: 21167: 21166: 21157: 21146: 21145: 21144: 21140: 21136: 21132: 21110: 21106: 21092: 21091: 21090: 21080: 21076: 21062: 21061: 21060: 21058: 21054: 21049: 21046: 21045: 21038:will lead to a 21036:ratio estimator 21028: 21027: 20986: 20982: 20975: 20969: 20964: 20958: 20937: 20933: 20926: 20920: 20915: 20909: 20898: 20894: 20888: 20883: 20877: 20866: 20862: 20856: 20851: 20845: 20843: 20831: 20824: 20820: 20808: 20803: 20789: 20780: 20775: 20761: 20752: 20741: 20740: 20739: 20730: 20719: 20718: 20717: 20713: 20709: 20708: 20704: 20681: 20676: 20670: 20661: 20657: 20656: 20652: 20639: 20634: 20628: 20619: 20615: 20614: 20610: 20608: 20593: 20589: 20582: 20572: 20568: 20561: 20559: 20558: 20554: 20546: 20543: 20542: 20511: 20507: 20500: 20494: 20489: 20483: 20462: 20458: 20451: 20445: 20440: 20434: 20423: 20419: 20413: 20408: 20402: 20391: 20387: 20381: 20376: 20370: 20368: 20356: 20349: 20345: 20333: 20328: 20314: 20305: 20300: 20286: 20277: 20266: 20265: 20264: 20255: 20244: 20243: 20242: 20237: 20234: 20233: 20189: 20185: 20178: 20172: 20167: 20161: 20140: 20136: 20129: 20123: 20118: 20112: 20101: 20097: 20091: 20086: 20080: 20069: 20065: 20059: 20054: 20048: 20036: 20031: 20018: 20013: 19996: 19984: 19980: 19971: 19967: 19963: 19959: 19944: 19939: 19925: 19916: 19911: 19897: 19888: 19877: 19876: 19875: 19866: 19855: 19854: 19853: 19848: 19845: 19844: 19808: 19803: 19797: 19794: 19793: 19776: 19771: 19765: 19762: 19761: 19744: 19733: 19732: 19731: 19722: 19711: 19710: 19709: 19707: 19704: 19703: 19697: 19658: 19654: 19647: 19641: 19636: 19630: 19609: 19605: 19598: 19592: 19587: 19581: 19570: 19566: 19560: 19555: 19549: 19538: 19534: 19528: 19523: 19517: 19515: 19503: 19496: 19492: 19480: 19475: 19461: 19452: 19447: 19433: 19424: 19413: 19412: 19411: 19402: 19391: 19390: 19389: 19385: 19381: 19380: 19376: 19353: 19348: 19342: 19333: 19329: 19328: 19324: 19311: 19306: 19300: 19291: 19287: 19286: 19282: 19280: 19265: 19261: 19254: 19244: 19240: 19233: 19231: 19230: 19226: 19218: 19215: 19214: 19188: 19183: 19170: 19165: 19148: 19136: 19132: 19123: 19119: 19115: 19111: 19093: 19088: 19082: 19073: 19069: 19068: 19064: 19051: 19046: 19040: 19031: 19027: 19026: 19022: 19020: 19005: 19001: 18994: 18984: 18980: 18973: 18971: 18969: 18966: 18965: 18919: 18914: 18908: 18894: 18889: 18883: 18881: 18869: 18862: 18858: 18849: 18838: 18837: 18836: 18827: 18816: 18815: 18814: 18813: 18809: 18808: 18804: 18790: 18786: 18777: 18773: 18772: 18768: 18757: 18754: 18753: 18731: 18727: 18725: 18722: 18721: 18704: 18700: 18698: 18695: 18694: 18674: 18669: 18656: 18652: 18647: 18644: 18643: 18623: 18618: 18605: 18601: 18596: 18593: 18592: 18585: 18557: 18538: 18534: 18532: 18529: 18528: 18505: 18498: 18494: 18492: 18489: 18488: 18486: 18484: 18483: 18481: 18435: 18429: 18425: 18423: 18409: 18405: 18403: 18401: 18389: 18382: 18378: 18364: 18360: 18358: 18344: 18343: 18342: 18338: 18337: 18333: 18312: 18308: 18306: 18299: 18295: 18279: 18275: 18267: 18264: 18263: 18228: 18222: 18218: 18216: 18202: 18198: 18196: 18194: 18182: 18175: 18171: 18157: 18153: 18151: 18137: 18136: 18134: 18131: 18130: 18104: 18100: 18098: 18090: 18087: 18086: 18044: 18038: 18034: 18032: 18018: 18014: 18012: 17998: 17994: 17992: 17985: 17981: 17967: 17966: 17955: 17951: 17949: 17935: 17934: 17932: 17929: 17928: 17927:, we get that: 17883: 17877: 17873: 17871: 17857: 17853: 17851: 17850: 17846: 17832: 17831: 17820: 17816: 17814: 17812: 17809: 17808: 17779: 17772: 17768: 17764: 17762: 17753: 17749: 17748: 17744: 17730: 17729: 17723: 17719: 17717: 17714: 17713: 17680: 17676: 17674: 17671: 17670: 17642: 17638: 17636: 17629: 17625: 17608: 17607: 17605: 17602: 17601: 17579: 17578: 17576: 17573: 17572: 17548: 17544: 17542: 17535: 17531: 17514: 17511: 17510: 17464: 17458: 17454: 17452: 17438: 17434: 17432: 17430: 17418: 17411: 17407: 17393: 17389: 17387: 17373: 17372: 17371: 17367: 17366: 17362: 17333: 17330: 17329: 17299: 17292: 17276: 17275: 17266: 17262: 17255: 17251: 17250: 17244: 17239: 17237: 17228: 17224: 17208: 17204: 17192: 17187: 17185: 17171: 17170: 17168: 17165: 17164: 17141: 17138: 17137: 17130: 17106: 17101: 17097: 17093: 17087: 17076: 17075: 17074: 17062: 17058: 17056: 17053: 17052: 17032: 17028: 17022: 17018: 17010: 17002: 17000: 16999: 16993: 16982: 16981: 16980: 16975: 16972: 16971: 16954: 16950: 16941: 16937: 16935: 16932: 16931: 16901: 16896: 16885: 16884: 16873: 16871: 16868: 16867: 16839: 16837: 16836: 16816: 16815: 16810: 16807: 16806: 16790: 16787: 16786: 16779: 16751: 16747: 16741: 16737: 16729: 16721: 16719: 16718: 16712: 16708: 16693: 16689: 16683: 16679: 16667: 16663: 16654: 16650: 16644: 16640: 16632: 16626: 16622: 16617: 16614: 16613: 16593: 16589: 16584: 16576: 16574: 16573: 16567: 16563: 16548: 16544: 16535: 16531: 16522: 16518: 16513: 16507: 16503: 16498: 16495: 16494: 16448: 16445: 16444: 16404: 16401: 16400: 16393: 16380: 16374: 16345: 16334: 16333: 16332: 16327: 16320: 16309: 16308: 16307: 16306: 16299: 16295: 16280: 16276: 16259: 16248: 16247: 16246: 16241: 16235: 16224: 16223: 16222: 16205: 16203: 16199: 16185: 16182: 16181: 16165: 16162: 16161: 16145: 16142: 16141: 16113: 16110: 16109: 16081: 16078: 16077: 16044: 16043: 16041: 16038: 16037: 16020: 16016: 16001: 15997: 15988: 15984: 15982: 15979: 15978: 15958: 15947: 15946: 15945: 15943: 15940: 15939: 15915: 15912: 15911: 15895: 15892: 15891: 15875: 15872: 15871: 15839: 15823: 15822: 15813: 15809: 15802: 15798: 15797: 15791: 15786: 15784: 15775: 15764: 15763: 15762: 15746: 15742: 15730: 15725: 15723: 15709: 15708: 15706: 15703: 15702: 15682: 15678: 15657: 15653: 15638: 15634: 15626: 15623: 15622: 15606: 15603: 15602: 15586: 15583: 15582: 15565: 15561: 15559: 15556: 15555: 15539: 15536: 15535: 15502: 15498: 15477: 15473: 15458: 15454: 15427: 15423: 15414: 15410: 15398: 15379: 15375: 15360: 15356: 15347: 15343: 15323: 15320: 15319: 15299: 15295: 15280: 15279: 15277: 15274: 15273: 15257: 15254: 15253: 15230: 15226: 15205: 15201: 15193: 15189: 15184: 15178: 15167: 15140: 15137: 15136: 15113: 15108: 15075: 15052: 15041: 15037: 15022: 15018: 15016: 15012: 15011: 15010: 15006: 15005: 14976: 14973: 14972: 14918: 14915: 14914: 14894: 14890: 14867: 14864: 14863: 14823: 14820: 14819: 14757: 14754: 14753: 14695: 14692: 14691: 14654: 14651: 14650: 14624: 14621: 14620: 14600: 14596: 14573: 14570: 14569: 14552: 14551: 14543: 14537: 14532: 14515: 14511: 14507: 14494: 14488: 14484: 14481: 14480: 14470: 14464: 14459: 14446: 14442: 14441: 14437: 14427: 14421: 14416: 14403: 14399: 14398: 14394: 14388: 14384: 14378: 14374: 14362: 14358: 14346: 14335: 14330: 14321: 14306: 14302: 14284: 14280: 14265: 14261: 14255: 14251: 14239: 14235: 14223: 14212: 14207: 14198: 14188: 14182: 14177: 14170: 14169: 14159: 14153: 14148: 14135: 14131: 14130: 14126: 14120: 14104: 14100: 14085: 14067: 14063: 14057: 14052: 14048: 14035: 14029: 14025: 14021: 14019: 14016: 14015: 13991: 13990: 13976: 13971: 13965: 13951: 13947: 13945: 13929: 13925: 13924: 13920: 13916: 13912: 13901: 13895: 13891: 13888: 13887: 13864: 13860: 13852: 13848: 13847: 13843: 13836: 13827: 13823: 13819: 13815: 13811: 13809: 13792: 13788: 13787: 13783: 13779: 13775: 13764: 13758: 13753: 13745: 13743: 13740: 13739: 13716: 13712: 13710: 13707: 13706: 13689: 13685: 13683: 13680: 13679: 13665: 13664: 13650: 13645: 13639: 13621: 13615: 13610: 13597: 13593: 13592: 13588: 13584: 13580: 13569: 13563: 13559: 13556: 13555: 13532: 13528: 13518: 13512: 13507: 13494: 13490: 13489: 13485: 13478: 13460: 13455: 13450: 13446: 13435: 13430: 13417: 13413: 13409: 13405: 13401: 13399: 13398: 13394: 13383: 13377: 13372: 13364: 13362: 13359: 13358: 13341: 13338: 13337: 13321: 13318: 13317: 13300: 13296: 13290: 13279: 13267: 13264: 13263: 13247: 13244: 13243: 13227: 13224: 13223: 13203: 13198: 13185: 13181: 13163: 13159: 13157: 13154: 13153: 13130: 13126: 13111: 13110: 13090: 13087: 13086: 13066: 13062: 13039: 13036: 13035: 13010: 13006: 12971: 12968: 12967: 12943: 12939: 12924: 12923: 12915: 12912: 12911: 12907: 12866: 12862: 12860: 12852: 12849: 12848: 12829: 12826: 12825: 12824:and arithmetic 12809: 12806: 12805: 12789: 12786: 12785: 12771: 12735: 12731: 12720: 12717: 12716: 12696: 12692: 12690: 12687: 12686: 12667: 12655: 12651: 12632: 12623: 12619: 12617: 12614: 12613: 12590: 12586: 12563: 12560: 12559: 12537: 12534: 12533: 12516: 12512: 12510: 12507: 12506: 12490: 12487: 12486: 12483: 12477: 12451: 12450: 12444: 12439: 12429: 12418: 12402: 12398: 12392: 12381: 12370: 12369: 12354: 12350: 12344: 12333: 12320: 12317: 12316: 12300: 12297: 12296: 12276: 12271: 12258: 12254: 12236: 12232: 12230: 12227: 12226: 12207: 12204: 12203: 12186: 12181: 12168: 12163: 12150: 12146: 12144: 12141: 12140: 12124: 12121: 12120: 12103: 12099: 12090: 12086: 12078: 12075: 12074: 12057: 12053: 12044: 12040: 12032: 12029: 12028: 12011: 12007: 12005: 12002: 12001: 11984: 11980: 11978: 11975: 11974: 11967: 11940: 11937: 11936: 11916: 11912: 11906: 11902: 11872: 11868: 11866: 11863: 11862: 11842: 11838: 11815: 11812: 11811: 11786: 11782: 11746: 11744: 11741: 11740: 11720: 11716: 11693: 11690: 11689: 11688:Reciprocal: If 11665: 11662: 11661: 11641: 11637: 11596: 11593: 11592: 11572: 11568: 11545: 11542: 11541: 11537: 11509: 11504: 11495: 11489: 11484: 11457: 11454: 11453: 11429: 11424: 11415: 11409: 11404: 11392: 11387: 11381: 11377: 11376: 11372: 11358: 11355: 11354: 11310: 11306: 11297: 11293: 11282: 11279: 11278: 11239: 11228: 11201: 11197: 11195: 11192: 11191: 11169: 11142: 11138: 11136: 11133: 11132: 11112: 11108: 11099: 11095: 11060: 11057: 11056: 11049: 11014: 11009: 11000: 10994: 10989: 10966: 10959: 10953: 10952: 10951: 10942: 10937: 10928: 10922: 10917: 10900: 10896: 10894: 10870: 10869: 10868: 10866: 10862: 10858: 10836: 10831: 10822: 10816: 10811: 10800: 10796: 10782: 10778: 10773: 10760: 10756: 10755: 10745: 10741: 10740: 10726: 10723: 10722: 10718: 10714: 10710: 10706: 10672: 10668: 10656: 10652: 10648: 10636: 10621: 10617: 10616: 10614: 10610: 10606: 10587: 10578: 10574: 10561: 10556: 10543: 10539: 10538: 10530: 10516: 10513: 10512: 10508: 10500: 10472: 10468: 10440: 10436: 10432: 10416: 10405: 10399: 10388: 10380: 10366: 10363: 10362: 10322: 10318: 10302: 10290: 10275: 10271: 10270: 10268: 10264: 10260: 10241: 10224: 10220: 10206: 10202: 10197: 10183: 10175: 10161: 10158: 10157: 10122: 10118: 10114: 10102: 10087: 10083: 10082: 10080: 10076: 10072: 10053: 10046: 10041: 10030: 10022: 10008: 10005: 10004: 9968: 9961: 9957: 9938: 9936: 9932: 9928: 9909: 9902: 9898: 9893: 9882: 9874: 9860: 9857: 9856: 9825: 9821: 9817: 9810: 9806: 9787: 9785: 9781: 9777: 9758: 9751: 9746: 9735: 9727: 9713: 9710: 9709: 9662: 9658: 9649: 9645: 9643: 9640: 9639: 9617: 9614: 9613: 9610: 9591: 9590: 9562: 9558: 9548: 9546: 9542: 9511: 9507: 9497: 9495: 9491: 9487: 9470: 9466: 9448: 9444: 9434: 9432: 9428: 9406: 9402: 9384: 9380: 9370: 9368: 9364: 9360: 9358: 9342: 9338: 9336: 9329: 9325: 9318: 9306: 9302: 9293: 9289: 9265: 9264: 9229: 9227: 9223: 9213: 9178: 9174: 9167: 9165: 9161: 9157: 9155: 9139: 9135: 9133: 9126: 9122: 9115: 9088: 9087: 9052: 9050: 9046: 9042: 9025: 9021: 8996: 8994: 8990: 8986: 8984: 8968: 8964: 8962: 8955: 8951: 8944: 8916: 8914: 8911: 8910: 8891: 8888: 8887: 8871: 8868: 8867: 8864: 8823: 8820: 8819: 8777: 8773: 8766: 8764: 8760: 8747: 8743: 8731: 8724: 8720: 8683: 8679: 8670: 8665: 8644: 8641: 8640: 8564: 8561: 8560: 8505: 8501: 8492: 8487: 8466: 8463: 8462: 8443: 8440: 8439: 8423: 8420: 8419: 8416: 8388: 8384: 8375: 8371: 8351: 8348: 8347: 8315: 8311: 8309: 8306: 8305: 8271: 8267: 8266: 8262: 8256: 8252: 8243: 8239: 8219: 8215: 8205: 8201: 8183: 8179: 8177: 8174: 8173: 8154: 8151: 8150: 8122: 8119: 8118: 8115:log-transformed 8089: 8085: 8078: 8074: 8054: 8051: 8050: 8046:, we get that: 8022: 8008: 8005: 8004: 7973: 7961: 7957: 7940: 7939: 7922: 7918: 7905: 7888: 7886: 7879: 7875: 7853: 7849: 7836: 7826: 7822: 7812: 7810: 7806: 7793: 7787: 7783: 7780: 7779: 7763: 7759: 7721: 7717: 7704: 7702: 7698: 7673: 7669: 7652: 7648: 7635: 7633: 7629: 7616: 7609: 7607: 7604: 7603: 7594: 7588: 7587:The parameters 7551: 7547: 7546: 7542: 7540: 7520: 7517: 7516: 7482: 7465: 7462: 7460: 7457: 7456: 7428: 7425: 7424: 7420:The arithmetic 7404: 7403: 7384: 7380: 7379: 7375: 7373: 7365: 7361: 7349: 7342: 7338: 7319: 7315: 7314: 7310: 7308: 7271: 7264: 7246: 7245: 7225: 7221: 7220: 7216: 7205: 7201: 7191: 7187: 7167: 7163: 7162: 7158: 7149: 7145: 7118: 7114: 7090: 7086: 7070: 7052: 7051: 7040: 7036: 7023: 7019: 7012: 7003: 6999: 6987: 6986: 6975: 6971: 6959: 6952: 6948: 6941: 6922: 6920: 6917: 6916: 6907: 6882: 6878: 6872: 6868: 6858: 6848: 6844: 6832: 6828: 6817: 6814: 6813: 6804: 6794: 6788: 6785: 6749: 6745: 6735: 6731: 6727: 6725: 6722: 6721: 6685: 6654: 6650: 6649: 6645: 6643: 6634: 6630: 6619: 6615: 6605: 6598: 6594: 6574: 6571: 6570: 6536: 6533: 6532: 6505: 6501: 6481: 6478: 6477: 6454: 6450: 6449: 6445: 6425: 6422: 6421: 6401: 6397: 6388: 6384: 6364: 6361: 6360: 6339: 6335: 6326: 6322: 6302: 6299: 6298: 6291: 6279: 6266: 6238: 6235: 6234: 6211: 6207: 6199: 6196: 6195: 6167: 6164: 6163: 6126: 6123: 6122: 6088: 6085: 6084: 6068: 6065: 6064: 6057: 6036: 6033: 6032: 6012: 6009: 6008: 5982: 5978: 5972: 5968: 5931: 5927: 5923: 5913: 5909: 5903: 5899: 5869: 5865: 5859: 5855: 5837: 5833: 5832: 5830: 5826: 5822: 5815: 5813: 5796: 5793: 5792: 5773: 5770: 5769: 5744: 5741: 5740: 5709: 5703: 5699: 5693: 5689: 5679: 5675: 5665: 5658: 5654: 5644: 5642: 5636: 5625: 5619: 5616: 5615: 5598: 5592: 5566: 5562: 5551: 5548: 5547: 5525: 5522: 5521: 5498: 5494: 5483: 5480: 5479: 5451: 5447: 5416: 5413: 5405: 5402: 5401: 5377: 5371: 5367: 5361: 5357: 5347: 5343: 5331: 5327: 5316: 5313: 5312: 5306: 5260: 5256: 5255: 5251: 5234: 5230: 5218: 5214: 5201: 5192: 5188: 5179: 5175: 5174: 5170: 5158: 5154: 5149: 5138: 5135: 5134: 5099: 5095: 5085: 5076: 5072: 5071: 5067: 5058: 5054: 5049: 5038: 5035: 5034: 5015: 5013: 5010: 5009: 4993: 4991: 4988: 4987: 4967: 4963: 4945: 4941: 4939: 4936: 4935: 4912: 4903: 4894: 4893: 4885: 4883: 4880: 4879: 4876: 4862: 4861: 4832: 4828: 4811: 4809: 4805: 4801: 4785: 4763: 4759: 4742: 4740: 4736: 4723: 4719: 4709: 4707: 4704: 4703: 4653: 4652: 4647: 4644: 4643: 4621: 4618: 4617: 4571: 4569: 4565: 4544: 4540: 4538: 4535: 4534: 4524: 4505: 4504: 4484: 4480: 4473: 4463: 4459: 4437: 4435: 4431: 4427: 4399: 4386: 4381: 4372: 4371: 4352: 4347: 4315: 4313: 4309: 4303: 4294: 4293: 4261: 4259: 4255: 4243: 4242: 4241: 4236: 4234: 4208: 4206: 4202: 4196: 4187: 4186: 4154: 4152: 4148: 4141: 4129: 4128: 4127: 4122: 4120: 4111: 4110: 4104: 4103: 4070: 4069: 4059: 4058: 4053: 4052: 4051: 4036: 4035: 4034: 4029: 4027: 4018: 4017: 4011: 4010: 3989: 3988: 3978: 3977: 3972: 3971: 3970: 3955: 3954: 3953: 3948: 3946: 3939: 3924: 3920: 3916: 3914: 3911: 3910: 3862: 3861: 3856: 3853: 3852: 3830: 3827: 3826: 3804: 3801: 3800: 3777: 3773: 3758: 3757: 3737: 3734: 3733: 3708: 3704: 3699: 3696: 3695: 3679: 3676: 3675: 3653: 3650: 3649: 3621: 3617: 3607: 3603: 3586: 3583: 3582: 3560: 3557: 3556: 3553: 3528: 3524: 3519: 3516: 3515: 3487: 3483: 3478: 3475: 3474: 3454: 3450: 3441: 3437: 3432: 3429: 3428: 3408: 3404: 3395: 3391: 3386: 3383: 3382: 3350: 3345: 3331: 3326: 3318: 3316: 3309: 3305: 3290: 3286: 3281: 3278: 3277: 3242: 3237: 3224: 3219: 3213: 3209: 3203: 3198: 3192: 3188: 3171: 3168: 3167: 3144: 3139: 3130: 3127: 3126: 3109: 3105: 3103: 3100: 3099: 3068: 3065: 3064: 3041: 3037: 3032: 3029: 3028: 3008: 3004: 2999: 2996: 2995: 2958: 2955: 2954: 2922: 2918: 2913: 2910: 2909: 2877: 2873: 2868: 2865: 2864: 2840: 2836: 2825: 2822: 2821: 2784: 2781: 2780: 2751: 2748: 2747: 2711: 2708: 2707: 2691: 2688: 2687: 2658: 2654: 2646: 2643: 2642: 2617: 2614: 2613: 2597: 2594: 2593: 2577: 2574: 2573: 2547: 2544: 2543: 2540: 2535: 2529:are specified. 2522: 2518: 2440: 2439: 2435: 2427: 2415: 2414: 2410: 2399:random variable 2332: 2328: 2300: 2295: 2282: 2278: 2276: 2269: 2265: 2257: 2254: 2253: 2252: 2251: 2196: 2192: 2173: 2169: 2164: 2163: 2159: 2143: 2141: 2128: 2124: 2122: 2115: 2111: 2101: 2096: 2093: 2092: 2041: 2037: 2022: 2021: 2017: 1991: 1989: 1988: 1984: 1976: 1965: 1962: 1961: 1960: 1959: 1910: 1906: 1891: 1890: 1886: 1860: 1858: 1856: 1852: 1830: 1829: 1828: 1826: 1822: 1805: 1802: 1801: 1772: 1771: 1759: 1755: 1751: 1746: 1744: 1738: 1737: 1732: 1720: 1716: 1712: 1707: 1700: 1699: 1694: 1691: 1690: 1679: 1677: 1654: 1648: 1644: 1638: 1634: 1621: 1617: 1607: 1597: 1593: 1577: 1575: 1569: 1558: 1549: 1546: 1545: 1533: 1488: 1481: 1477: 1455: 1451: 1447: 1438: 1434: 1429: 1426: 1425: 1386: 1382: 1378: 1374: 1348: 1344: 1337: 1333: 1307: 1303: 1296: 1292: 1275: 1272: 1271: 1267:Excess kurtosis 1234: 1230: 1219: 1195: 1191: 1187: 1177: 1173: 1168: 1165: 1164: 1131: 1127: 1114: 1110: 1078: 1074: 1061: 1057: 1052: 1049: 1048: 1012: 1008: 998: 994: 983: 980: 979: 931: 928: 927: 886: 882: 880: 870: 866: 855: 852: 851: 811: 807: 784: 781: 780: 779: 778: 730: 726: 718: 714: 709: 702: 698: 687: 684: 683: 632: 630: 626: 598: 594: 571: 569: 565: 552: 548: 532: 530: 525: 522: 521: 488: 484: 480: 474: 450: 446: 445: 443: 439: 435: 407: 397: 392: 387: 384: 383: 326: 323: 322: 281: 278: 277: 276: 220: 217: 216: 180: 176: 165: 161: 150: 147: 146: 111: 108: 107: 105: 99: 77: 74: 73: 51: 48: 47: 44: 38: 28: 23: 22: 15: 12: 11: 5: 28186: 28176: 28175: 28170: 28165: 28160: 28155: 28138: 28137: 28135: 28134: 28124: 28113: 28110: 28109: 28107: 28106: 28101: 28096: 28091: 28086: 28081: 28079:Location–scale 28076: 28071: 28066: 28061: 28056: 28050: 28048: 28044: 28043: 28041: 28040: 28035: 28028: 28023: 28015: 28013: 28002: 28001: 27999: 27998: 27993: 27988: 27981: 27976: 27969: 27964: 27957: 27952: 27947: 27942: 27940:Wrapped Cauchy 27937: 27935:Wrapped normal 27932: 27927: 27922: 27911: 27909: 27903: 27902: 27900: 27899: 27898: 27897: 27892: 27890:Normal-inverse 27887: 27882: 27872: 27871: 27870: 27860: 27852: 27847: 27842: 27833: 27832: 27831: 27821: 27813: 27808: 27803: 27798: 27797: 27796: 27786: 27779: 27778: 27777: 27772: 27762: 27757: 27749: 27747: 27739: 27738: 27735: 27734: 27732: 27731: 27725: 27723: 27714: 27708: 27707: 27704: 27703: 27701: 27700: 27695: 27690: 27682: 27674: 27666: 27657: 27648: 27639: 27630: 27621: 27616: 27611: 27606: 27600: 27598: 27592: 27591: 27589: 27588: 27583: 27581:Variance-gamma 27578: 27573: 27565: 27560: 27555: 27550: 27545: 27540: 27532: 27527: 27526: 27525: 27515: 27510: 27505: 27499: 27494: 27489: 27484: 27479: 27474: 27469: 27461: 27456: 27448: 27443: 27437: 27435: 27427: 27426: 27424: 27423: 27421:Wilks's lambda 27418: 27417: 27416: 27406: 27401: 27396: 27391: 27386: 27381: 27376: 27371: 27366: 27361: 27359:Mittag-Leffler 27356: 27351: 27346: 27341: 27336: 27331: 27326: 27321: 27316: 27311: 27306: 27301: 27300: 27299: 27289: 27280: 27275: 27270: 27269: 27268: 27258: 27256:gamma/Gompertz 27253: 27252: 27251: 27246: 27236: 27231: 27226: 27225: 27224: 27212: 27211: 27210: 27205: 27200: 27190: 27189: 27188: 27178: 27173: 27168: 27167: 27166: 27165: 27164: 27154: 27144: 27139: 27134: 27129: 27124: 27119: 27113: 27111: 27108:semi-infinite 27103: 27102: 27100: 27099: 27094: 27089: 27084: 27079: 27074: 27069: 27064: 27059: 27054: 27049: 27044: 27039: 27034: 27029: 27024: 27019: 27014: 27008: 27006: 26997: 26991: 26990: 26987: 26986: 26984: 26983: 26978: 26973: 26968: 26963: 26958: 26953: 26948: 26943: 26938: 26933: 26928: 26923: 26918: 26913: 26908: 26903: 26898: 26892: 26890: 26887:with infinite 26884: 26883: 26881: 26880: 26875: 26870: 26865: 26860: 26855: 26850: 26849: 26848: 26841:Hypergeometric 26838: 26833: 26828: 26823: 26818: 26812: 26810: 26801: 26795: 26794: 26782: 26781: 26774: 26767: 26759: 26753: 26752: 26745: 26744:External links 26742: 26741: 26740: 26715: 26694: 26682:(5): 341–352. 26667: 26660: 26639: 26624: 26621: 26619: 26618: 26609: 26600: 26579: 26538: 26485: 26438: 26416: 26409: 26391: 26372:(2): 221–230. 26352: 26313: 26254: 26225:(4): 963–971. 26209: 26196: 26189: 26169: 26156:10.1086/260062 26134: 26101: 26084: 26073: 26066: 26035: 25988: 25931: 25882: 25831: 25774: 25717: 25702: 25675:(1): 171–178. 25655: 25636:(3): 245–250. 25620: 25591:(1): 119–121. 25585:Bioinformatics 25571: 25528: 25515: 25500: 25482: 25457: 25440: 25420: 25383:(4): 264–278. 25360: 25314: 25287: 25244: 25169: 25150:(3): 241–249. 25134: 25090: 25079: 25070: 25061: 25049: 25029: 25018:(6): 441–444. 25002: 24963:Johnson, N. L. 24954: 24943:(2): 505–515. 24927: 24906: 24899: 24863: 24836: 24802: 24766: 24747:(3): 121–148. 24731: 24720:(7): 946–950. 24704: 24677: 24665: 24656: 24637:(2): 141–151. 24621: 24602:(2): 789–805. 24586: 24573:(5): 341–352. 24553: 24544: 24532: 24497:Bioinformatics 24483: 24462: 24445:|journal= 24417: 24391: 24342: 24318: 24264: 24251:(5): 341–352. 24231: 24201: 24137: 24122: 24109:(3): 327–347. 24086: 24042: 24023:(3): 211–216. 24007: 23977: 23962: 23942: 23936: 23908: 23881: 23854: 23796: 23782: 23751: 23719: 23687: 23618: 23616: 23613: 23610: 23609: 23591: 23588: 23585: 23580: 23576: 23572: 23569: 23563: 23558: 23554: 23548: 23542: 23539: 23536: 23531: 23527: 23523: 23520: 23514: 23509: 23505: 23499: 23492: 23488: 23482: 23477: 23473: 23467: 23460: 23456: 23450: 23445: 23441: 23434: 23428: 23423: 23418: 23413: 23409: 23405: 23400: 23395: 23391: 23387: 23382: 23379: 23374: 23371: 23366: 23359: 23356: 23349: 23344: 23337: 23334: 23327: 23323: 23319: 23315: 23308: 23304: 23297: 23294: 23291: 23286: 23282: 23278: 23275: 23269: 23264: 23260: 23254: 23248: 23245: 23242: 23237: 23233: 23229: 23226: 23220: 23215: 23211: 23205: 23198: 23194: 23188: 23183: 23179: 23173: 23166: 23162: 23156: 23151: 23147: 23140: 23134: 23128: 23122: 23117: 23113: 23109: 23103: 23100: 23092: 23087: 23083: 23079: 23073: 23070: 23061: 23057: 23051: 23047: 23041: 23036: 23032: 23028: 23025: 23020: 23015: 23011: 23007: 23004: 22998: 22978: 22963: 22959: 22955: 22950: 22945: 22941: 22937: 22934: 22931: 22928: 22924: 22920: 22917: 22912: 22905: 22902: 22895: 22892: 22870: 22867: 22864: 22859: 22855: 22851: 22847: 22843: 22840: 22837: 22834: 22831: 22828: 22825: 22822: 22818: 22814: 22811: 22808: 22803: 22799: 22795: 22792: 22789: 22786: 22782: 22778: 22775: 22770: 22763: 22760: 22753: 22750: 22728: 22721: 22718: 22692: 22689: 22686: 22683: 22678: 22671: 22668: 22661: 22657: 22653: 22648: 22641: 22638: 22631: 22628: 22625: 22622: 22619: 22614: 22607: 22604: 22597: 22594: 22591: 22588: 22584: 22580: 22577: 22572: 22568: 22564: 22561: 22558: 22538: 22535: 22532: 22529: 22524: 22520: 22516: 22512: 22508: 22505: 22502: 22499: 22496: 22493: 22490: 22487: 22482: 22478: 22474: 22471: 22468: 22465: 22462: 22459: 22453: 22450: 22444: 22439: 22432: 22429: 22422: 22419: 22416: 22413: 22393: 22390: 22386: 22382: 22379: 22376: 22373: 22370: 22367: 22362: 22358: 22354: 22349: 22345: 22341: 22338: 22332: 22329: 22321: 22314: 22311: 22286: 22276: 22275: 22273: 22270: 22266: 22265: 22260: 22255: 22250: 22243: 22240: 22239: 22238: 22231: 22223: 22220: 22217: 22202: 22195: 22184: 22175: 22172: 22171: 22170: 22167: 22157: 22154:scientometrics 22150: 22110: 22098: 22095: 22092: 22089: 22069: 22066: 22062: 22056: 22052: 22047: 22043: 22040: 22037: 22034: 22023:error function 22010: 21989: 21984: 21981: 21976: 21972: 21969: 21966: 21963: 21939: 21928: 21911: 21908: 21907: 21906: 21905: 21904: 21890: 21889: 21888: 21887: 21869: 21868: 21859: 21856: 21844: 21843: 21840: 21837: 21825: 21822: 21821: 21820: 21817: 21810: 21807: 21803: 21790: 21781: 21778: 21771: 21768: 21765: 21757: 21754: 21751: 21746: 21743: 21742: 21741: 21738: 21733:The length of 21731: 21728: 21723: 21722:Human behavior 21720: 21694: 21691: 21676: 21669: 21664: 21661: 21658: 21635: 21615: 21612: 21609: 21606: 21580: 21576: 21571: 21568: 21548: 21536: 21530: 21515: 21508: 21505: 21502: 21497: 21493: 21489: 21486: 21480: 21475: 21471: 21465: 21459: 21456: 21453: 21448: 21444: 21440: 21437: 21431: 21426: 21422: 21416: 21409: 21405: 21399: 21394: 21390: 21384: 21377: 21373: 21367: 21362: 21358: 21351: 21345: 21342: 21337: 21334: 21329: 21324: 21320: 21316: 21311: 21306: 21302: 21298: 21293: 21290: 21285: 21282: 21277: 21273: 21269: 21264: 21260: 21256: 21252: 21248: 21244: 21239: 21234: 21229: 21225: 21221: 21216: 21211: 21207: 21203: 21198: 21195: 21190: 21187: 21182: 21175: 21172: 21165: 21160: 21153: 21150: 21143: 21139: 21135: 21131: 21128: 21124: 21118: 21113: 21109: 21105: 21099: 21096: 21088: 21083: 21079: 21075: 21069: 21066: 21057: 21053: 21009: 21000: 20997: 20994: 20989: 20985: 20981: 20978: 20972: 20967: 20963: 20957: 20951: 20948: 20945: 20940: 20936: 20932: 20929: 20923: 20918: 20914: 20908: 20901: 20897: 20891: 20886: 20882: 20876: 20869: 20865: 20859: 20854: 20850: 20838: 20835: 20830: 20827: 20823: 20819: 20816: 20811: 20806: 20802: 20796: 20793: 20788: 20783: 20778: 20774: 20768: 20765: 20760: 20755: 20748: 20745: 20738: 20733: 20726: 20723: 20716: 20712: 20707: 20703: 20699: 20689: 20684: 20679: 20675: 20669: 20664: 20660: 20655: 20647: 20642: 20637: 20633: 20627: 20622: 20618: 20613: 20607: 20601: 20596: 20592: 20588: 20585: 20580: 20575: 20571: 20567: 20564: 20557: 20553: 20550: 20525: 20522: 20519: 20514: 20510: 20506: 20503: 20497: 20492: 20488: 20482: 20476: 20473: 20470: 20465: 20461: 20457: 20454: 20448: 20443: 20439: 20433: 20426: 20422: 20416: 20411: 20407: 20401: 20394: 20390: 20384: 20379: 20375: 20363: 20360: 20355: 20352: 20348: 20344: 20341: 20336: 20331: 20327: 20321: 20318: 20313: 20308: 20303: 20299: 20293: 20290: 20285: 20280: 20273: 20270: 20263: 20258: 20251: 20248: 20241: 20210: 20203: 20200: 20197: 20192: 20188: 20184: 20181: 20175: 20170: 20166: 20160: 20154: 20151: 20148: 20143: 20139: 20135: 20132: 20126: 20121: 20117: 20111: 20104: 20100: 20094: 20089: 20085: 20079: 20072: 20068: 20062: 20057: 20053: 20047: 20044: 20039: 20034: 20030: 20026: 20021: 20016: 20012: 20008: 20003: 20000: 19995: 19992: 19987: 19983: 19979: 19974: 19970: 19966: 19962: 19958: 19955: 19952: 19947: 19942: 19938: 19932: 19929: 19924: 19919: 19914: 19910: 19904: 19901: 19896: 19891: 19884: 19881: 19874: 19869: 19862: 19859: 19852: 19811: 19806: 19802: 19779: 19774: 19770: 19747: 19740: 19737: 19730: 19725: 19718: 19715: 19700: 19699: 19681: 19672: 19669: 19666: 19661: 19657: 19653: 19650: 19644: 19639: 19635: 19629: 19623: 19620: 19617: 19612: 19608: 19604: 19601: 19595: 19590: 19586: 19580: 19573: 19569: 19563: 19558: 19554: 19548: 19541: 19537: 19531: 19526: 19522: 19510: 19507: 19502: 19499: 19495: 19491: 19488: 19483: 19478: 19474: 19468: 19465: 19460: 19455: 19450: 19446: 19440: 19437: 19432: 19427: 19420: 19417: 19410: 19405: 19398: 19395: 19388: 19384: 19379: 19375: 19371: 19361: 19356: 19351: 19347: 19341: 19336: 19332: 19327: 19319: 19314: 19309: 19305: 19299: 19294: 19290: 19285: 19279: 19273: 19268: 19264: 19260: 19257: 19252: 19247: 19243: 19239: 19236: 19229: 19225: 19222: 19196: 19191: 19186: 19182: 19178: 19173: 19168: 19164: 19160: 19155: 19152: 19147: 19144: 19139: 19135: 19131: 19126: 19122: 19118: 19114: 19110: 19101: 19096: 19091: 19087: 19081: 19076: 19072: 19067: 19059: 19054: 19049: 19045: 19039: 19034: 19030: 19025: 19019: 19013: 19008: 19004: 19000: 18997: 18992: 18987: 18983: 18979: 18976: 18935: 18927: 18922: 18917: 18913: 18907: 18902: 18897: 18892: 18888: 18876: 18873: 18868: 18865: 18861: 18857: 18852: 18845: 18842: 18835: 18830: 18823: 18820: 18812: 18807: 18803: 18800: 18793: 18789: 18785: 18780: 18776: 18771: 18767: 18764: 18761: 18747:respectively. 18734: 18730: 18707: 18703: 18682: 18677: 18672: 18668: 18664: 18659: 18655: 18651: 18631: 18626: 18621: 18617: 18613: 18608: 18604: 18600: 18584: 18581: 18564: 18561: 18556: 18553: 18550: 18547: 18544: 18541: 18537: 18512: 18509: 18504: 18501: 18497: 18462: 18453: 18450: 18447: 18444: 18441: 18438: 18432: 18428: 18422: 18417: 18412: 18408: 18396: 18393: 18388: 18385: 18381: 18377: 18372: 18367: 18363: 18357: 18351: 18348: 18341: 18336: 18332: 18328: 18320: 18315: 18311: 18305: 18302: 18298: 18294: 18291: 18288: 18285: 18282: 18278: 18274: 18271: 18246: 18243: 18240: 18237: 18234: 18231: 18225: 18221: 18215: 18210: 18205: 18201: 18189: 18186: 18181: 18178: 18174: 18170: 18165: 18160: 18156: 18150: 18144: 18141: 18112: 18107: 18103: 18097: 18094: 18069: 18062: 18059: 18056: 18053: 18050: 18047: 18041: 18037: 18031: 18026: 18021: 18017: 18011: 18006: 18001: 17997: 17991: 17988: 17984: 17980: 17974: 17971: 17963: 17958: 17954: 17948: 17942: 17939: 17908: 17901: 17898: 17895: 17892: 17889: 17886: 17880: 17876: 17870: 17865: 17860: 17856: 17849: 17845: 17839: 17836: 17828: 17823: 17819: 17795: 17788: 17785: 17782: 17775: 17771: 17767: 17761: 17756: 17752: 17747: 17743: 17737: 17734: 17726: 17722: 17683: 17679: 17656: 17650: 17645: 17641: 17635: 17632: 17628: 17624: 17621: 17615: 17612: 17586: 17583: 17556: 17551: 17547: 17541: 17538: 17534: 17530: 17527: 17524: 17521: 17518: 17507: 17506: 17491: 17482: 17479: 17476: 17473: 17470: 17467: 17461: 17457: 17451: 17446: 17441: 17437: 17425: 17422: 17417: 17414: 17410: 17406: 17401: 17396: 17392: 17386: 17380: 17377: 17370: 17365: 17361: 17358: 17355: 17352: 17349: 17346: 17343: 17340: 17337: 17308: 17305: 17302: 17295: 17290: 17283: 17280: 17274: 17269: 17265: 17261: 17258: 17254: 17247: 17243: 17236: 17231: 17227: 17222: 17217: 17211: 17207: 17203: 17200: 17195: 17191: 17184: 17178: 17175: 17145: 17129: 17126: 17110: 17104: 17100: 17096: 17090: 17083: 17080: 17073: 17070: 17065: 17061: 17040: 17035: 17031: 17025: 17021: 17017: 17013: 17005: 16996: 16989: 16986: 16979: 16957: 16953: 16949: 16944: 16940: 16924:t distribution 16905: 16899: 16892: 16889: 16883: 16879: 16876: 16855: 16849: 16845: 16842: 16835: 16832: 16829: 16823: 16820: 16814: 16794: 16778: 16772: 16759: 16754: 16750: 16744: 16740: 16736: 16732: 16724: 16715: 16711: 16707: 16704: 16701: 16696: 16692: 16686: 16682: 16678: 16675: 16670: 16666: 16662: 16657: 16653: 16647: 16643: 16639: 16635: 16629: 16625: 16621: 16601: 16596: 16592: 16587: 16579: 16570: 16566: 16562: 16559: 16556: 16551: 16547: 16543: 16538: 16534: 16530: 16525: 16521: 16516: 16510: 16506: 16502: 16482: 16479: 16476: 16473: 16470: 16467: 16464: 16461: 16458: 16455: 16452: 16432: 16429: 16426: 16423: 16420: 16417: 16414: 16411: 16408: 16392: 16389: 16373: 16370: 16358: 16354: 16348: 16341: 16338: 16330: 16323: 16316: 16313: 16305: 16302: 16298: 16294: 16291: 16288: 16283: 16279: 16274: 16270: 16262: 16255: 16252: 16244: 16238: 16231: 16228: 16221: 16218: 16212: 16209: 16202: 16198: 16195: 16192: 16189: 16169: 16149: 16129: 16126: 16123: 16120: 16117: 16097: 16094: 16091: 16088: 16085: 16051: 16048: 16023: 16019: 16015: 16012: 16009: 16004: 16000: 15996: 15991: 15987: 15961: 15954: 15951: 15919: 15899: 15879: 15853: 15848: 15842: 15837: 15830: 15827: 15821: 15816: 15812: 15808: 15805: 15801: 15794: 15790: 15783: 15778: 15771: 15768: 15760: 15755: 15749: 15745: 15741: 15738: 15733: 15729: 15722: 15716: 15713: 15690: 15685: 15681: 15677: 15674: 15671: 15668: 15665: 15660: 15656: 15652: 15649: 15646: 15641: 15637: 15633: 15630: 15610: 15590: 15568: 15564: 15543: 15513: 15510: 15505: 15501: 15497: 15494: 15491: 15488: 15485: 15480: 15476: 15472: 15469: 15466: 15461: 15457: 15453: 15450: 15447: 15444: 15441: 15438: 15435: 15430: 15426: 15422: 15417: 15413: 15409: 15406: 15401: 15397: 15393: 15390: 15387: 15382: 15378: 15374: 15371: 15368: 15363: 15359: 15355: 15350: 15346: 15342: 15339: 15336: 15333: 15330: 15327: 15307: 15302: 15298: 15294: 15291: 15288: 15283: 15261: 15241: 15238: 15233: 15229: 15225: 15222: 15219: 15214: 15211: 15208: 15204: 15196: 15192: 15188: 15181: 15176: 15173: 15170: 15166: 15162: 15159: 15156: 15153: 15150: 15147: 15144: 15129:same procedure 15112: 15109: 15107: 15104: 15103: 15102: 15086: 15081: 15078: 15073: 15069: 15066: 15061: 15056: 15051: 15048: 15044: 15040: 15035: 15030: 15025: 15021: 15015: 15009: 15004: 15001: 14998: 14995: 14992: 14989: 14986: 14983: 14980: 14962: 14946: 14943: 14940: 14937: 14934: 14931: 14928: 14925: 14922: 14902: 14897: 14893: 14889: 14886: 14883: 14880: 14877: 14874: 14871: 14851: 14848: 14845: 14842: 14839: 14836: 14833: 14830: 14827: 14816: 14809: 14797: 14794: 14791: 14788: 14785: 14782: 14779: 14776: 14773: 14770: 14767: 14764: 14761: 14741: 14738: 14735: 14732: 14729: 14726: 14723: 14720: 14717: 14714: 14711: 14708: 14705: 14702: 14699: 14679: 14676: 14673: 14670: 14667: 14664: 14661: 14658: 14634: 14631: 14628: 14608: 14603: 14599: 14595: 14592: 14589: 14586: 14583: 14580: 14577: 14550: 14546: 14540: 14535: 14531: 14527: 14523: 14518: 14514: 14510: 14506: 14503: 14500: 14497: 14495: 14491: 14487: 14483: 14482: 14477: 14473: 14467: 14462: 14458: 14454: 14449: 14445: 14440: 14434: 14430: 14424: 14419: 14415: 14411: 14406: 14402: 14397: 14391: 14387: 14381: 14377: 14373: 14368: 14365: 14361: 14355: 14352: 14349: 14345: 14338: 14333: 14329: 14324: 14320: 14317: 14314: 14309: 14305: 14301: 14298: 14295: 14292: 14287: 14283: 14279: 14276: 14273: 14268: 14264: 14258: 14254: 14250: 14245: 14242: 14238: 14232: 14229: 14226: 14222: 14215: 14210: 14206: 14201: 14197: 14194: 14191: 14189: 14185: 14180: 14176: 14172: 14171: 14166: 14162: 14156: 14151: 14147: 14143: 14138: 14134: 14129: 14123: 14119: 14115: 14112: 14107: 14103: 14099: 14096: 14093: 14088: 14084: 14080: 14076: 14070: 14066: 14060: 14056: 14051: 14047: 14044: 14041: 14038: 14036: 14032: 14028: 14024: 14023: 14005: 14004: 13989: 13984: 13979: 13974: 13970: 13964: 13959: 13954: 13950: 13944: 13940: 13932: 13928: 13923: 13919: 13915: 13910: 13907: 13904: 13902: 13898: 13894: 13890: 13889: 13886: 13882: 13878: 13875: 13867: 13863: 13855: 13851: 13846: 13842: 13839: 13830: 13826: 13822: 13818: 13814: 13808: 13805: 13802: 13795: 13791: 13786: 13782: 13778: 13773: 13770: 13767: 13765: 13761: 13756: 13752: 13748: 13747: 13727: 13724: 13719: 13715: 13692: 13688: 13663: 13658: 13653: 13648: 13644: 13638: 13634: 13628: 13624: 13618: 13613: 13609: 13605: 13600: 13596: 13591: 13587: 13583: 13578: 13575: 13572: 13570: 13566: 13562: 13558: 13557: 13554: 13550: 13546: 13543: 13535: 13531: 13525: 13521: 13515: 13510: 13506: 13502: 13497: 13493: 13488: 13484: 13481: 13476: 13473: 13470: 13463: 13458: 13454: 13449: 13445: 13438: 13433: 13429: 13425: 13420: 13416: 13412: 13408: 13404: 13397: 13392: 13389: 13386: 13384: 13380: 13375: 13371: 13367: 13366: 13345: 13325: 13303: 13299: 13293: 13288: 13285: 13282: 13278: 13274: 13271: 13251: 13231: 13211: 13206: 13201: 13197: 13193: 13188: 13184: 13180: 13177: 13174: 13171: 13166: 13162: 13150: 13138: 13133: 13129: 13125: 13122: 13119: 13114: 13109: 13106: 13103: 13100: 13097: 13094: 13074: 13069: 13065: 13061: 13058: 13055: 13052: 13049: 13046: 13043: 13032: 13021: 13018: 13013: 13009: 13005: 13002: 12999: 12996: 12993: 12990: 12987: 12984: 12981: 12978: 12975: 12951: 12946: 12942: 12938: 12935: 12932: 12927: 12922: 12919: 12906: 12903: 12891: 12890: 12879: 12874: 12869: 12865: 12859: 12856: 12833: 12813: 12793: 12770: 12767: 12743: 12738: 12734: 12730: 12727: 12724: 12699: 12695: 12674: 12670: 12666: 12663: 12658: 12654: 12650: 12647: 12644: 12641: 12631: 12626: 12622: 12601: 12598: 12593: 12589: 12585: 12582: 12579: 12576: 12573: 12570: 12567: 12547: 12544: 12541: 12519: 12515: 12494: 12476: 12473: 12459: 12454: 12447: 12442: 12438: 12432: 12427: 12424: 12421: 12417: 12410: 12405: 12401: 12395: 12390: 12387: 12384: 12380: 12373: 12368: 12365: 12362: 12357: 12353: 12347: 12342: 12339: 12336: 12332: 12327: 12324: 12304: 12284: 12279: 12274: 12270: 12266: 12261: 12257: 12253: 12250: 12247: 12244: 12239: 12235: 12211: 12189: 12184: 12180: 12176: 12171: 12166: 12162: 12158: 12153: 12149: 12128: 12106: 12102: 12098: 12093: 12089: 12085: 12082: 12060: 12056: 12052: 12047: 12043: 12039: 12036: 12014: 12010: 11987: 11983: 11966: 11963: 11962: 11961: 11950: 11947: 11944: 11924: 11919: 11915: 11909: 11905: 11898: 11895: 11892: 11889: 11886: 11883: 11880: 11875: 11871: 11850: 11845: 11841: 11837: 11834: 11831: 11828: 11825: 11822: 11819: 11808: 11797: 11794: 11789: 11785: 11778: 11775: 11772: 11769: 11766: 11763: 11760: 11754: 11751: 11728: 11723: 11719: 11715: 11712: 11709: 11706: 11703: 11700: 11697: 11686: 11675: 11672: 11669: 11649: 11644: 11640: 11633: 11630: 11627: 11624: 11621: 11618: 11615: 11612: 11609: 11606: 11603: 11600: 11580: 11575: 11571: 11567: 11564: 11561: 11558: 11555: 11552: 11549: 11536: 11533: 11517: 11512: 11507: 11503: 11498: 11492: 11487: 11483: 11479: 11476: 11473: 11470: 11467: 11464: 11461: 11440: 11432: 11427: 11423: 11418: 11412: 11407: 11403: 11399: 11396: 11390: 11384: 11380: 11375: 11371: 11368: 11365: 11362: 11342: 11339: 11336: 11333: 11330: 11327: 11324: 11321: 11318: 11313: 11309: 11305: 11300: 11296: 11292: 11289: 11286: 11261: 11258: 11255: 11252: 11249: 11246: 11243: 11238: 11235: 11231: 11227: 11224: 11221: 11218: 11215: 11212: 11209: 11204: 11200: 11179: 11176: 11172: 11168: 11165: 11162: 11159: 11156: 11153: 11150: 11145: 11141: 11120: 11115: 11111: 11107: 11102: 11098: 11094: 11091: 11088: 11085: 11082: 11079: 11076: 11073: 11070: 11067: 11064: 11048: 11045: 11044: 11043: 11042: 11041: 11029: 11022: 11017: 11012: 11008: 11003: 10997: 10992: 10988: 10984: 10981: 10978: 10975: 10972: 10969: 10962: 10956: 10945: 10940: 10936: 10931: 10925: 10920: 10916: 10912: 10909: 10903: 10899: 10893: 10890: 10887: 10884: 10881: 10878: 10873: 10865: 10861: 10857: 10854: 10845: 10839: 10834: 10830: 10825: 10819: 10814: 10810: 10806: 10803: 10799: 10795: 10792: 10789: 10786: 10781: 10777: 10772: 10769: 10763: 10759: 10754: 10748: 10744: 10739: 10736: 10733: 10730: 10716: 10712: 10711:) with mean, μ 10708: 10704: 10701: 10700: 10699: 10687: 10680: 10675: 10671: 10667: 10664: 10659: 10655: 10651: 10646: 10643: 10639: 10635: 10632: 10629: 10624: 10620: 10613: 10609: 10605: 10602: 10594: 10591: 10586: 10581: 10577: 10573: 10570: 10567: 10564: 10560: 10555: 10552: 10546: 10542: 10537: 10533: 10529: 10526: 10523: 10520: 10506: 10498: 10497:LogNormal6(m,σ 10495: 10494: 10493: 10481: 10475: 10471: 10467: 10464: 10461: 10458: 10455: 10452: 10447: 10444: 10439: 10435: 10431: 10428: 10423: 10420: 10411: 10408: 10404: 10398: 10395: 10391: 10387: 10383: 10379: 10376: 10373: 10370: 10357: 10356: 10355: 10343: 10336: 10333: 10330: 10325: 10321: 10317: 10314: 10311: 10308: 10305: 10300: 10297: 10293: 10289: 10286: 10283: 10278: 10274: 10267: 10263: 10259: 10256: 10248: 10245: 10238: 10235: 10232: 10227: 10223: 10219: 10216: 10213: 10210: 10205: 10201: 10196: 10193: 10189: 10186: 10182: 10178: 10174: 10171: 10168: 10165: 10148: 10147: 10146: 10134: 10125: 10121: 10117: 10112: 10109: 10105: 10101: 10098: 10095: 10090: 10086: 10079: 10075: 10071: 10068: 10060: 10057: 10052: 10049: 10045: 10040: 10037: 10033: 10029: 10025: 10021: 10018: 10015: 10012: 9995: 9994: 9993: 9981: 9974: 9971: 9964: 9960: 9956: 9953: 9950: 9947: 9944: 9941: 9935: 9931: 9927: 9924: 9916: 9913: 9906: 9901: 9897: 9892: 9889: 9885: 9881: 9877: 9873: 9870: 9867: 9864: 9851: 9850: 9849: 9837: 9828: 9824: 9820: 9813: 9809: 9805: 9802: 9799: 9796: 9793: 9790: 9784: 9780: 9776: 9773: 9765: 9762: 9757: 9754: 9750: 9745: 9742: 9738: 9734: 9730: 9726: 9723: 9720: 9717: 9665: 9661: 9657: 9652: 9648: 9627: 9624: 9621: 9609: 9606: 9605: 9604: 9585: 9580: 9576: 9573: 9570: 9565: 9561: 9557: 9554: 9551: 9545: 9541: 9538: 9534: 9529: 9525: 9522: 9519: 9514: 9510: 9506: 9503: 9500: 9494: 9490: 9484: 9479: 9473: 9469: 9465: 9462: 9459: 9456: 9451: 9447: 9443: 9440: 9437: 9431: 9427: 9424: 9420: 9415: 9409: 9405: 9401: 9398: 9395: 9392: 9387: 9383: 9379: 9376: 9373: 9367: 9363: 9357: 9350: 9345: 9341: 9335: 9332: 9328: 9324: 9321: 9319: 9317: 9314: 9309: 9305: 9301: 9296: 9292: 9288: 9285: 9282: 9279: 9276: 9273: 9270: 9267: 9266: 9259: 9254: 9250: 9247: 9244: 9241: 9238: 9235: 9232: 9226: 9222: 9219: 9216: 9210: 9205: 9201: 9198: 9195: 9192: 9189: 9186: 9181: 9177: 9173: 9170: 9164: 9160: 9154: 9147: 9142: 9138: 9132: 9129: 9125: 9121: 9118: 9116: 9114: 9111: 9108: 9105: 9102: 9099: 9096: 9093: 9090: 9089: 9082: 9077: 9073: 9070: 9067: 9064: 9061: 9058: 9055: 9049: 9045: 9039: 9034: 9028: 9024: 9020: 9017: 9014: 9011: 9008: 9005: 9002: 8999: 8993: 8989: 8983: 8976: 8971: 8967: 8961: 8958: 8954: 8950: 8947: 8945: 8943: 8940: 8937: 8934: 8931: 8928: 8925: 8922: 8919: 8918: 8895: 8875: 8863: 8860: 8827: 8816: 8815: 8803: 8798: 8794: 8791: 8788: 8785: 8780: 8776: 8772: 8769: 8763: 8758: 8750: 8746: 8739: 8736: 8730: 8727: 8723: 8719: 8716: 8713: 8709: 8706: 8703: 8700: 8697: 8694: 8691: 8686: 8682: 8678: 8673: 8668: 8664: 8660: 8657: 8654: 8651: 8648: 8625: 8622: 8619: 8616: 8613: 8610: 8607: 8604: 8601: 8598: 8595: 8592: 8589: 8586: 8583: 8580: 8577: 8574: 8571: 8568: 8553: 8552: 8541: 8538: 8535: 8531: 8528: 8525: 8522: 8519: 8516: 8513: 8508: 8504: 8500: 8495: 8490: 8486: 8482: 8479: 8476: 8473: 8470: 8458:is defined as 8447: 8427: 8415: 8412: 8411: 8410: 8399: 8391: 8387: 8383: 8378: 8374: 8370: 8367: 8364: 8361: 8358: 8355: 8329: 8326: 8323: 8318: 8314: 8302: 8301: 8290: 8285: 8282: 8279: 8274: 8270: 8265: 8259: 8255: 8251: 8246: 8242: 8238: 8233: 8230: 8227: 8222: 8218: 8214: 8211: 8208: 8204: 8200: 8197: 8194: 8191: 8186: 8182: 8158: 8138: 8135: 8132: 8129: 8126: 8111: 8110: 8099: 8092: 8088: 8084: 8081: 8077: 8073: 8070: 8067: 8064: 8061: 8058: 8035: 8032: 8028: 8025: 8021: 8018: 8015: 8012: 7979:Comparison of 7972: 7969: 7954: 7953: 7938: 7934: 7925: 7921: 7917: 7914: 7911: 7908: 7903: 7900: 7897: 7894: 7891: 7885: 7882: 7878: 7874: 7871: 7868: 7864: 7856: 7852: 7848: 7845: 7842: 7839: 7834: 7829: 7825: 7821: 7818: 7815: 7809: 7805: 7802: 7799: 7796: 7794: 7790: 7786: 7782: 7781: 7778: 7774: 7766: 7762: 7758: 7755: 7752: 7749: 7746: 7743: 7740: 7737: 7734: 7731: 7724: 7720: 7716: 7713: 7710: 7707: 7701: 7697: 7694: 7691: 7687: 7681: 7676: 7672: 7668: 7665: 7662: 7655: 7651: 7647: 7644: 7641: 7638: 7632: 7628: 7625: 7622: 7619: 7617: 7615: 7612: 7611: 7581: 7580: 7569: 7564: 7561: 7554: 7550: 7545: 7539: 7536: 7533: 7530: 7527: 7524: 7497: 7494: 7491: 7488: 7485: 7480: 7477: 7474: 7471: 7468: 7444: 7441: 7438: 7435: 7432: 7418: 7417: 7402: 7397: 7394: 7387: 7383: 7378: 7368: 7364: 7357: 7354: 7348: 7345: 7341: 7337: 7332: 7329: 7322: 7318: 7313: 7307: 7304: 7301: 7298: 7295: 7292: 7287: 7284: 7281: 7278: 7275: 7270: 7267: 7265: 7263: 7260: 7257: 7254: 7251: 7248: 7247: 7244: 7241: 7238: 7235: 7228: 7224: 7219: 7215: 7208: 7204: 7200: 7197: 7194: 7190: 7186: 7183: 7180: 7177: 7170: 7166: 7161: 7157: 7152: 7148: 7144: 7141: 7138: 7135: 7132: 7129: 7126: 7121: 7117: 7113: 7110: 7107: 7104: 7101: 7098: 7093: 7089: 7085: 7082: 7079: 7076: 7073: 7071: 7069: 7066: 7063: 7060: 7057: 7054: 7053: 7050: 7043: 7039: 7035: 7032: 7029: 7026: 7022: 7018: 7015: 7013: 7011: 7006: 7002: 6998: 6995: 6992: 6989: 6988: 6985: 6978: 6974: 6967: 6964: 6958: 6955: 6951: 6947: 6944: 6942: 6940: 6937: 6934: 6931: 6928: 6925: 6924: 6904: 6903: 6892: 6885: 6881: 6875: 6871: 6865: 6862: 6857: 6854: 6851: 6847: 6843: 6840: 6835: 6831: 6827: 6824: 6821: 6784: 6781: 6752: 6748: 6742: 6739: 6734: 6730: 6718: 6717: 6706: 6701: 6698: 6695: 6692: 6689: 6684: 6681: 6678: 6675: 6672: 6669: 6666: 6657: 6653: 6648: 6642: 6637: 6633: 6629: 6622: 6618: 6612: 6609: 6604: 6601: 6597: 6593: 6590: 6587: 6584: 6581: 6578: 6540: 6516: 6513: 6508: 6504: 6500: 6497: 6494: 6491: 6488: 6485: 6457: 6453: 6448: 6444: 6441: 6438: 6435: 6432: 6429: 6404: 6400: 6396: 6391: 6387: 6383: 6380: 6377: 6374: 6371: 6368: 6342: 6338: 6334: 6329: 6325: 6321: 6318: 6315: 6312: 6309: 6306: 6290: 6287: 6278: 6275: 6265: 6262: 6248: 6245: 6242: 6222: 6219: 6214: 6210: 6206: 6203: 6183: 6180: 6177: 6174: 6171: 6151: 6148: 6145: 6142: 6139: 6136: 6133: 6130: 6110: 6107: 6104: 6101: 6098: 6095: 6092: 6072: 6056: 6053: 6040: 6016: 6005: 6004: 5990: 5985: 5981: 5975: 5971: 5967: 5964: 5961: 5958: 5955: 5952: 5949: 5943: 5934: 5930: 5926: 5921: 5916: 5912: 5906: 5902: 5898: 5895: 5892: 5889: 5886: 5883: 5880: 5877: 5872: 5868: 5862: 5858: 5854: 5851: 5848: 5845: 5840: 5836: 5829: 5825: 5821: 5818: 5812: 5809: 5806: 5803: 5800: 5777: 5757: 5754: 5751: 5748: 5730: 5729: 5716: 5712: 5706: 5702: 5696: 5692: 5688: 5685: 5682: 5678: 5671: 5668: 5661: 5657: 5653: 5650: 5647: 5639: 5634: 5631: 5628: 5624: 5580: 5575: 5572: 5569: 5565: 5561: 5558: 5555: 5529: 5509: 5504: 5501: 5497: 5493: 5490: 5487: 5463: 5459: 5454: 5450: 5446: 5443: 5440: 5437: 5434: 5431: 5428: 5425: 5422: 5419: 5412: 5409: 5398: 5397: 5384: 5380: 5374: 5370: 5364: 5360: 5356: 5353: 5350: 5346: 5342: 5339: 5334: 5330: 5326: 5323: 5320: 5305: 5302: 5294: 5293: 5282: 5279: 5276: 5273: 5266: 5263: 5259: 5254: 5250: 5245: 5240: 5237: 5233: 5229: 5224: 5221: 5217: 5213: 5208: 5205: 5200: 5195: 5191: 5187: 5182: 5178: 5173: 5169: 5164: 5161: 5157: 5152: 5148: 5145: 5142: 5124: 5123: 5112: 5105: 5102: 5098: 5092: 5089: 5084: 5079: 5075: 5070: 5066: 5061: 5057: 5052: 5048: 5045: 5042: 5018: 5008:. The mean of 4996: 4975: 4970: 4966: 4962: 4959: 4956: 4953: 4948: 4944: 4919: 4915: 4910: 4906: 4902: 4897: 4892: 4888: 4875: 4872: 4858: 4857: 4845: 4836: 4831: 4826: 4823: 4820: 4817: 4814: 4808: 4804: 4800: 4797: 4792: 4789: 4784: 4780: 4775: 4767: 4762: 4757: 4754: 4751: 4748: 4745: 4739: 4735: 4732: 4729: 4726: 4722: 4716: 4713: 4682: 4679: 4673: 4670: 4664: 4661: 4656: 4628: 4614: 4613: 4601: 4596: 4592: 4589: 4586: 4583: 4580: 4577: 4574: 4568: 4564: 4561: 4558: 4555: 4552: 4547: 4543: 4523: 4520: 4519: 4518: 4503: 4496: 4487: 4483: 4476: 4466: 4462: 4458: 4455: 4452: 4449: 4446: 4443: 4434: 4430: 4426: 4423: 4409: 4403: 4398: 4392: 4385: 4380: 4377: 4375: 4373: 4364: 4358: 4351: 4345: 4340: 4333: 4330: 4327: 4324: 4321: 4312: 4306: 4302: 4299: 4297: 4295: 4291: 4286: 4279: 4276: 4273: 4270: 4267: 4258: 4251: 4246: 4239: 4232: 4227: 4223: 4220: 4217: 4214: 4211: 4205: 4199: 4195: 4192: 4190: 4188: 4184: 4179: 4172: 4169: 4166: 4163: 4160: 4151: 4144: 4137: 4132: 4125: 4119: 4116: 4114: 4112: 4107: 4099: 4096: 4093: 4090: 4087: 4084: 4081: 4073: 4062: 4056: 4044: 4039: 4032: 4026: 4023: 4021: 4019: 4014: 4006: 4003: 4000: 3992: 3981: 3975: 3963: 3958: 3951: 3945: 3942: 3940: 3938: 3935: 3932: 3927: 3923: 3919: 3918: 3888: 3882: 3879: 3876: 3870: 3865: 3837: 3811: 3797: 3796: 3785: 3780: 3776: 3772: 3769: 3766: 3761: 3756: 3753: 3750: 3747: 3744: 3741: 3719: 3711: 3707: 3683: 3660: 3633: 3624: 3620: 3616: 3613: 3606: 3602: 3599: 3596: 3593: 3567: 3552: 3549: 3531: 3527: 3490: 3486: 3457: 3453: 3449: 3444: 3440: 3411: 3407: 3403: 3398: 3394: 3368: 3361: 3353: 3348: 3344: 3334: 3329: 3325: 3315: 3312: 3308: 3304: 3301: 3298: 3293: 3289: 3261: 3245: 3240: 3236: 3232: 3227: 3222: 3218: 3206: 3201: 3197: 3191: 3187: 3184: 3181: 3178: 3155: 3147: 3142: 3138: 3112: 3108: 3084: 3081: 3078: 3075: 3072: 3052: 3044: 3040: 3011: 3007: 2983: 2977: 2974: 2971: 2968: 2965: 2939: 2936: 2933: 2930: 2925: 2921: 2894: 2891: 2888: 2885: 2880: 2876: 2843: 2839: 2835: 2832: 2806: 2803: 2800: 2797: 2794: 2791: 2758: 2728:expected value 2715: 2695: 2684: 2683: 2670: 2667: 2664: 2661: 2657: 2653: 2650: 2627: 2624: 2621: 2601: 2581: 2554: 2539: 2536: 2534: 2531: 2477:Francis Galton 2373: 2372: 2361: 2358: 2355: 2352: 2349: 2346: 2343: 2338: 2335: 2331: 2327: 2324: 2321: 2318: 2315: 2309: 2306: 2303: 2299: 2290: 2285: 2281: 2275: 2272: 2268: 2264: 2237: 2229: 2225: 2222: 2219: 2216: 2213: 2210: 2207: 2202: 2199: 2195: 2191: 2177: 2168: 2162: 2158: 2155: 2152: 2149: 2136: 2131: 2127: 2121: 2118: 2114: 2108: 2105: 2089: 2083: 2082: 2065: 2058: 2055: 2044: 2040: 2036: 2033: 2030: 2025: 2009: 2006: 2003: 2000: 1997: 1987: 1983: 1980: 1975: 1972: 1948: 1941: 1927: 1924: 1913: 1909: 1905: 1902: 1899: 1894: 1878: 1875: 1872: 1869: 1866: 1847: 1844: 1841: 1838: 1833: 1825: 1821: 1818: 1815: 1812: 1799: 1793: 1792: 1776: 1762: 1758: 1750: 1745: 1743: 1740: 1739: 1736: 1733: 1723: 1719: 1711: 1706: 1705: 1703: 1688: 1682: 1681: 1661: 1657: 1651: 1647: 1641: 1637: 1633: 1630: 1627: 1620: 1613: 1610: 1600: 1596: 1592: 1586: 1583: 1572: 1567: 1564: 1561: 1557: 1542: 1536: 1535: 1530: 1524: 1523: 1508: 1496: 1493: 1487: 1484: 1480: 1473: 1462: 1459: 1450: 1446: 1441: 1437: 1423: 1417: 1416: 1402: 1399: 1395: 1389: 1385: 1381: 1377: 1373: 1370: 1364: 1361: 1357: 1351: 1347: 1340: 1336: 1332: 1329: 1323: 1320: 1316: 1310: 1306: 1299: 1295: 1291: 1288: 1282: 1269: 1263: 1262: 1248: 1245: 1242: 1237: 1233: 1229: 1226: 1223: 1217: 1210: 1207: 1203: 1198: 1194: 1190: 1186: 1183: 1176: 1162: 1156: 1155: 1140: 1134: 1130: 1126: 1123: 1117: 1113: 1109: 1106: 1099: 1092: 1089: 1086: 1081: 1077: 1073: 1070: 1067: 1060: 1046: 1040: 1039: 1024: 1015: 1011: 1007: 1004: 997: 993: 990: 977: 971: 970: 956: 950: 944: 941: 938: 925: 919: 918: 903: 894: 889: 885: 879: 876: 869: 865: 862: 849: 843: 842: 831: 828: 825: 822: 817: 814: 810: 806: 803: 800: 797: 794: 791: 788: 763: 759: 756: 753: 750: 747: 744: 741: 736: 733: 729: 721: 717: 713: 708: 705: 701: 697: 694: 681: 675: 674: 662: 657: 653: 650: 647: 644: 641: 638: 635: 629: 625: 622: 618: 613: 602: 597: 589: 586: 583: 580: 577: 568: 564: 561: 558: 555: 551: 545: 538: 519: 513: 512: 500: 491: 487: 483: 477: 472: 465: 462: 459: 456: 453: 449: 442: 438: 434: 431: 414: 411: 406: 403: 396: 381: 375: 374: 360: 354: 351: 348: 345: 339: 336: 333: 320: 314: 313: 308:(logarithm of 294: 291: 288: 271:(logarithm of 257: 251: 248: 245: 242: 239: 233: 230: 227: 214: 208: 207: 192: 183: 179: 174: 171: 164: 160: 157: 144: 140: 139: 124: 121: 118: 97: 94: 93: 81: 58: 36: 26: 9: 6: 4: 3: 2: 28185: 28174: 28171: 28169: 28166: 28164: 28161: 28159: 28156: 28154: 28151: 28150: 28148: 28133: 28125: 28123: 28115: 28114: 28111: 28105: 28102: 28100: 28097: 28095: 28092: 28090: 28087: 28085: 28082: 28080: 28077: 28075: 28072: 28070: 28067: 28065: 28062: 28060: 28057: 28055: 28052: 28051: 28049: 28045: 28039: 28036: 28033: 28029: 28027: 28024: 28021: 28017: 28016: 28014: 28012: 28007: 28003: 27997: 27994: 27992: 27989: 27986: 27982: 27980: 27977: 27974: 27970: 27968: 27965: 27962: 27958: 27956: 27953: 27951: 27948: 27946: 27943: 27941: 27938: 27936: 27933: 27931: 27928: 27926: 27923: 27920: 27919: 27913: 27912: 27910: 27908: 27904: 27896: 27893: 27891: 27888: 27886: 27883: 27881: 27878: 27877: 27876: 27873: 27869: 27866: 27865: 27864: 27861: 27859: 27858: 27853: 27851: 27850:Matrix normal 27848: 27846: 27843: 27840: 27839: 27834: 27830: 27827: 27826: 27825: 27822: 27820: 27819: 27816:Multivariate 27814: 27812: 27809: 27807: 27804: 27802: 27799: 27795: 27792: 27791: 27790: 27787: 27784: 27780: 27776: 27773: 27771: 27768: 27767: 27766: 27763: 27761: 27758: 27755: 27751: 27750: 27748: 27746: 27743:Multivariate 27740: 27730: 27727: 27726: 27724: 27718: 27715: 27709: 27699: 27696: 27694: 27691: 27689: 27687: 27683: 27681: 27679: 27675: 27673: 27671: 27667: 27665: 27663: 27658: 27656: 27654: 27649: 27647: 27645: 27640: 27638: 27636: 27631: 27629: 27627: 27622: 27620: 27617: 27615: 27612: 27610: 27607: 27605: 27602: 27601: 27599: 27595:with support 27593: 27587: 27584: 27582: 27579: 27577: 27574: 27572: 27571: 27566: 27564: 27561: 27559: 27556: 27554: 27551: 27549: 27546: 27544: 27541: 27539: 27538: 27533: 27531: 27528: 27524: 27521: 27520: 27519: 27516: 27514: 27511: 27509: 27508: 27500: 27498: 27495: 27493: 27490: 27488: 27485: 27483: 27480: 27478: 27475: 27473: 27470: 27468: 27467: 27462: 27460: 27457: 27455: 27454: 27449: 27447: 27444: 27442: 27439: 27438: 27436: 27432:on the whole 27428: 27422: 27419: 27415: 27412: 27411: 27410: 27407: 27405: 27404:type-2 Gumbel 27402: 27400: 27397: 27395: 27392: 27390: 27387: 27385: 27382: 27380: 27377: 27375: 27372: 27370: 27367: 27365: 27362: 27360: 27357: 27355: 27352: 27350: 27347: 27345: 27342: 27340: 27337: 27335: 27332: 27330: 27327: 27325: 27322: 27320: 27317: 27315: 27312: 27310: 27307: 27305: 27302: 27298: 27295: 27294: 27293: 27290: 27288: 27286: 27281: 27279: 27276: 27274: 27273:Half-logistic 27271: 27267: 27264: 27263: 27262: 27259: 27257: 27254: 27250: 27247: 27245: 27242: 27241: 27240: 27237: 27235: 27232: 27230: 27229:Folded normal 27227: 27223: 27220: 27219: 27218: 27217: 27213: 27209: 27206: 27204: 27201: 27199: 27196: 27195: 27194: 27191: 27187: 27184: 27183: 27182: 27179: 27177: 27174: 27172: 27169: 27163: 27160: 27159: 27158: 27155: 27153: 27150: 27149: 27148: 27145: 27143: 27140: 27138: 27135: 27133: 27130: 27128: 27125: 27123: 27120: 27118: 27115: 27114: 27112: 27104: 27098: 27095: 27093: 27090: 27088: 27085: 27083: 27080: 27078: 27075: 27073: 27072:Raised cosine 27070: 27068: 27065: 27063: 27060: 27058: 27055: 27053: 27050: 27048: 27045: 27043: 27040: 27038: 27035: 27033: 27030: 27028: 27025: 27023: 27020: 27018: 27015: 27013: 27010: 27009: 27007: 27001: 26998: 26992: 26982: 26979: 26977: 26974: 26972: 26969: 26967: 26964: 26962: 26959: 26957: 26954: 26952: 26949: 26947: 26946:Mixed Poisson 26944: 26942: 26939: 26937: 26934: 26932: 26929: 26927: 26924: 26922: 26919: 26917: 26914: 26912: 26909: 26907: 26904: 26902: 26899: 26897: 26894: 26893: 26891: 26885: 26879: 26876: 26874: 26871: 26869: 26866: 26864: 26861: 26859: 26856: 26854: 26851: 26847: 26844: 26843: 26842: 26839: 26837: 26834: 26832: 26829: 26827: 26826:Beta-binomial 26824: 26822: 26819: 26817: 26814: 26813: 26811: 26805: 26802: 26796: 26791: 26787: 26780: 26775: 26773: 26768: 26766: 26761: 26760: 26757: 26751: 26748: 26747: 26737: 26733: 26729: 26725: 26721: 26716: 26712: 26708: 26704: 26700: 26695: 26690: 26685: 26681: 26677: 26673: 26668: 26665: 26661: 26658: 26654: 26650: 26646: 26642: 26636: 26632: 26627: 26626: 26613: 26604: 26595: 26590: 26583: 26575: 26571: 26566: 26561: 26557: 26553: 26549: 26542: 26534: 26530: 26526: 26522: 26518: 26514: 26509: 26504: 26500: 26496: 26489: 26481: 26477: 26473: 26469: 26465: 26461: 26458:(80): 31–32. 26457: 26453: 26449: 26442: 26434: 26430: 26426: 26420: 26412: 26406: 26402: 26395: 26387: 26383: 26379: 26375: 26371: 26367: 26363: 26356: 26348: 26344: 26340: 26336: 26332: 26328: 26324: 26317: 26309: 26305: 26300: 26295: 26291: 26287: 26282: 26277: 26273: 26269: 26265: 26258: 26250: 26246: 26242: 26238: 26233: 26228: 26224: 26220: 26213: 26206: 26203:Bunchen, P., 26200: 26192: 26190:9780465043552 26186: 26182: 26181: 26173: 26165: 26161: 26157: 26153: 26149: 26145: 26138: 26130: 26126: 26121: 26116: 26112: 26105: 26098: 26094: 26088: 26082: 26077: 26069: 26063: 26059: 26055: 26054: 26046: 26039: 26031: 26027: 26023: 26019: 26015: 26011: 26007: 26003: 25999: 25992: 25984: 25980: 25975: 25970: 25966: 25962: 25958: 25954: 25951:(5): 053057. 25950: 25946: 25942: 25935: 25927: 25923: 25918: 25913: 25909: 25905: 25901: 25897: 25893: 25886: 25878: 25874: 25869: 25864: 25859: 25854: 25850: 25846: 25845:F1000Research 25842: 25835: 25827: 25823: 25818: 25813: 25809: 25805: 25801: 25797: 25793: 25789: 25785: 25778: 25770: 25766: 25761: 25756: 25752: 25748: 25744: 25740: 25736: 25732: 25728: 25721: 25713: 25706: 25698: 25694: 25690: 25686: 25682: 25678: 25674: 25670: 25666: 25659: 25651: 25647: 25643: 25639: 25635: 25631: 25624: 25616: 25612: 25608: 25604: 25599: 25594: 25590: 25586: 25582: 25575: 25567: 25563: 25559: 25555: 25551: 25547: 25543: 25539: 25532: 25525: 25519: 25511: 25507: 25503: 25497: 25493: 25486: 25471: 25467: 25461: 25453: 25452: 25444: 25436: 25429: 25427: 25425: 25416: 25412: 25407: 25402: 25398: 25394: 25390: 25386: 25382: 25378: 25374: 25367: 25365: 25356: 25352: 25347: 25342: 25338: 25334: 25330: 25323: 25321: 25319: 25310: 25306: 25302: 25298: 25291: 25283: 25279: 25275: 25271: 25267: 25263: 25259: 25255: 25248: 25240: 25236: 25231: 25226: 25222: 25218: 25214: 25210: 25206: 25202: 25197: 25192: 25188: 25184: 25180: 25173: 25165: 25161: 25157: 25153: 25149: 25145: 25138: 25129: 25124: 25120: 25116: 25112: 25108: 25104: 25097: 25095: 25088: 25083: 25074: 25065: 25059: 25053: 25047: 25044: 25038: 25036: 25034: 25025: 25021: 25017: 25013: 25006: 24998: 24994: 24990: 24986: 24982: 24978: 24974: 24970: 24969: 24964: 24958: 24950: 24946: 24942: 24938: 24931: 24922: 24917: 24910: 24902: 24896: 24892: 24888: 24883: 24878: 24874: 24867: 24859: 24855: 24851: 24847: 24840: 24832: 24828: 24824: 24820: 24813: 24806: 24797: 24792: 24788: 24784: 24780: 24773: 24771: 24762: 24758: 24754: 24750: 24746: 24742: 24735: 24727: 24723: 24719: 24715: 24708: 24700: 24696: 24692: 24688: 24681: 24675: 24669: 24660: 24652: 24648: 24644: 24640: 24636: 24632: 24625: 24617: 24613: 24609: 24605: 24601: 24597: 24590: 24581: 24576: 24572: 24568: 24564: 24557: 24548: 24539: 24537: 24528: 24524: 24519: 24514: 24510: 24506: 24502: 24498: 24494: 24487: 24472: 24466: 24460: 24459:9780470696750 24456: 24450: 24438: 24429: 24424: 24420: 24414: 24410: 24406: 24402: 24395: 24387: 24383: 24378: 24373: 24369: 24365: 24361: 24357: 24356:Ann Rheum Dis 24353: 24346: 24339: 24338:PharmaSUG2011 24335: 24331: 24328: 24322: 24314: 24310: 24305: 24300: 24295: 24290: 24286: 24282: 24278: 24271: 24269: 24259: 24254: 24250: 24246: 24242: 24235: 24227: 24223: 24219: 24215: 24208: 24206: 24197: 24193: 24188: 24183: 24179: 24175: 24170: 24165: 24161: 24157: 24153: 24146: 24144: 24142: 24135: 24132: 24126: 24117: 24112: 24108: 24104: 24097: 24090: 24081: 24076: 24072: 24068: 24064: 24060: 24053: 24046: 24038: 24034: 24030: 24026: 24022: 24018: 24011: 24003: 23999: 23995: 23991: 23984: 23982: 23973: 23969: 23965: 23959: 23955: 23954: 23946: 23939: 23933: 23928: 23923: 23919: 23912: 23901: 23894: 23893: 23885: 23874: 23867: 23866: 23858: 23844:on 2016-03-07 23840: 23836: 23832: 23827: 23822: 23818: 23814: 23807: 23800: 23793: 23789: 23785: 23779: 23775: 23771: 23764: 23762: 23760: 23758: 23756: 23740: 23736: 23732: 23726: 23724: 23709: 23705: 23698: 23696: 23694: 23692: 23673: 23669: 23665: 23661: 23657: 23652: 23647: 23643: 23639: 23638: 23630: 23623: 23619: 23586: 23583: 23578: 23574: 23567: 23561: 23556: 23552: 23546: 23537: 23534: 23529: 23525: 23518: 23512: 23507: 23503: 23497: 23490: 23486: 23480: 23475: 23471: 23465: 23458: 23454: 23448: 23443: 23439: 23432: 23426: 23416: 23411: 23407: 23403: 23398: 23393: 23389: 23380: 23377: 23372: 23364: 23357: 23354: 23347: 23342: 23335: 23332: 23321: 23317: 23313: 23306: 23302: 23292: 23289: 23284: 23280: 23273: 23267: 23262: 23258: 23252: 23243: 23240: 23235: 23231: 23224: 23218: 23213: 23209: 23203: 23196: 23192: 23186: 23181: 23177: 23171: 23164: 23160: 23154: 23149: 23145: 23138: 23132: 23126: 23115: 23111: 23101: 23098: 23085: 23081: 23071: 23068: 23059: 23055: 23049: 23045: 23034: 23030: 23023: 23013: 23009: 23002: 22996: 22982: 22961: 22957: 22953: 22943: 22939: 22932: 22929: 22926: 22922: 22918: 22910: 22900: 22890: 22865: 22862: 22857: 22853: 22845: 22838: 22835: 22832: 22829: 22823: 22820: 22816: 22812: 22809: 22801: 22797: 22790: 22787: 22784: 22780: 22776: 22768: 22758: 22748: 22726: 22716: 22684: 22681: 22676: 22666: 22655: 22646: 22636: 22629: 22626: 22620: 22612: 22602: 22592: 22589: 22586: 22582: 22578: 22570: 22566: 22559: 22556: 22530: 22527: 22522: 22518: 22510: 22503: 22500: 22497: 22494: 22488: 22480: 22476: 22469: 22466: 22463: 22457: 22451: 22448: 22437: 22427: 22417: 22414: 22411: 22388: 22384: 22377: 22374: 22371: 22368: 22360: 22356: 22352: 22347: 22343: 22336: 22330: 22327: 22319: 22309: 22296: 22290: 22281: 22277: 22269: 22264: 22261: 22259: 22256: 22254: 22251: 22249: 22246: 22245: 22236: 22232: 22228: 22224: 22221: 22218: 22215: 22211: 22207: 22203: 22200: 22196: 22193: 22189: 22185: 22182: 22178: 22177: 22168: 22165: 22161: 22158: 22155: 22151: 22148: 22144: 22140: 22136: 22132: 22128: 22123: 22119: 22115: 22111: 22093: 22067: 22064: 22060: 22054: 22050: 22045: 22038: 22035: 22032: 22024: 22008: 21987: 21982: 21979: 21974: 21970: 21967: 21964: 21961: 21953: 21937: 21929: 21926: 21922: 21918: 21914: 21913: 21902: 21899:as part of a 21898: 21894: 21893: 21892: 21891: 21885: 21882:based on the 21881: 21877: 21873: 21872: 21871: 21870: 21866: 21862: 21861: 21853: 21848: 21841: 21838: 21835: 21831: 21828: 21827: 21818: 21815: 21811: 21808: 21804: 21801: 21797: 21793: 21786: 21782: 21779: 21776: 21772: 21769: 21766: 21763: 21758: 21755: 21752: 21749: 21748: 21739: 21736: 21732: 21729: 21726: 21725: 21719: 21715: 21711: 21709: 21705: 21701: 21690: 21674: 21662: 21659: 21656: 21647: 21633: 21613: 21610: 21607: 21604: 21594: 21578: 21574: 21569: 21566: 21546: 21535: 21529: 21513: 21503: 21500: 21495: 21491: 21484: 21478: 21473: 21469: 21463: 21454: 21451: 21446: 21442: 21435: 21429: 21424: 21420: 21414: 21407: 21403: 21397: 21392: 21388: 21382: 21375: 21371: 21365: 21360: 21356: 21349: 21343: 21340: 21335: 21327: 21322: 21318: 21314: 21309: 21304: 21300: 21291: 21288: 21283: 21275: 21271: 21267: 21262: 21258: 21250: 21246: 21242: 21232: 21227: 21223: 21219: 21214: 21209: 21205: 21196: 21193: 21188: 21180: 21173: 21170: 21163: 21158: 21151: 21148: 21137: 21133: 21129: 21126: 21122: 21111: 21107: 21097: 21094: 21081: 21077: 21067: 21064: 21055: 21051: 21043: 21041: 21037: 21033: 21026: 21023: 21007: 20995: 20992: 20987: 20983: 20976: 20970: 20965: 20961: 20955: 20946: 20943: 20938: 20934: 20927: 20921: 20916: 20912: 20906: 20899: 20895: 20889: 20884: 20880: 20874: 20867: 20863: 20857: 20852: 20848: 20836: 20833: 20828: 20825: 20821: 20817: 20809: 20804: 20800: 20794: 20791: 20786: 20781: 20776: 20772: 20766: 20763: 20758: 20753: 20743: 20736: 20731: 20721: 20710: 20705: 20701: 20697: 20687: 20682: 20677: 20673: 20667: 20662: 20658: 20653: 20645: 20640: 20635: 20631: 20625: 20620: 20616: 20611: 20605: 20594: 20590: 20583: 20573: 20569: 20562: 20555: 20551: 20548: 20520: 20517: 20512: 20508: 20501: 20495: 20490: 20486: 20480: 20471: 20468: 20463: 20459: 20452: 20446: 20441: 20437: 20431: 20424: 20420: 20414: 20409: 20405: 20399: 20392: 20388: 20382: 20377: 20373: 20361: 20358: 20353: 20350: 20346: 20342: 20334: 20329: 20325: 20319: 20316: 20311: 20306: 20301: 20297: 20291: 20288: 20283: 20278: 20268: 20261: 20256: 20246: 20231: 20227: 20222: 20208: 20198: 20195: 20190: 20186: 20179: 20173: 20168: 20164: 20158: 20149: 20146: 20141: 20137: 20130: 20124: 20119: 20115: 20109: 20102: 20098: 20092: 20087: 20083: 20077: 20070: 20066: 20060: 20055: 20051: 20045: 20037: 20032: 20028: 20024: 20019: 20014: 20010: 20001: 19998: 19993: 19985: 19981: 19977: 19972: 19968: 19960: 19956: 19953: 19945: 19940: 19936: 19930: 19927: 19922: 19917: 19912: 19908: 19902: 19899: 19894: 19889: 19879: 19872: 19867: 19857: 19841: 19839: 19835: 19831: 19830:approximately 19827: 19809: 19804: 19800: 19777: 19772: 19768: 19745: 19735: 19728: 19723: 19713: 19698: 19695: 19679: 19667: 19664: 19659: 19655: 19648: 19642: 19637: 19633: 19627: 19618: 19615: 19610: 19606: 19599: 19593: 19588: 19584: 19578: 19571: 19567: 19561: 19556: 19552: 19546: 19539: 19535: 19529: 19524: 19520: 19508: 19505: 19500: 19497: 19493: 19489: 19481: 19476: 19472: 19466: 19463: 19458: 19453: 19448: 19444: 19438: 19435: 19430: 19425: 19415: 19408: 19403: 19393: 19382: 19377: 19373: 19369: 19359: 19354: 19349: 19345: 19339: 19334: 19330: 19325: 19317: 19312: 19307: 19303: 19297: 19292: 19288: 19283: 19277: 19266: 19262: 19255: 19245: 19241: 19234: 19227: 19223: 19220: 19212: 19209: 19189: 19184: 19180: 19176: 19171: 19166: 19162: 19153: 19150: 19145: 19137: 19133: 19129: 19124: 19120: 19112: 19108: 19099: 19094: 19089: 19085: 19079: 19074: 19070: 19065: 19057: 19052: 19047: 19043: 19037: 19032: 19028: 19023: 19017: 19006: 19002: 18995: 18985: 18981: 18974: 18963: 18960: 18958: 18954: 18953:relative-risk 18949: 18933: 18925: 18920: 18915: 18911: 18905: 18900: 18895: 18890: 18886: 18874: 18871: 18866: 18863: 18859: 18855: 18850: 18840: 18833: 18828: 18818: 18810: 18805: 18801: 18791: 18787: 18783: 18778: 18774: 18769: 18762: 18759: 18751: 18748: 18732: 18728: 18705: 18701: 18675: 18670: 18666: 18662: 18657: 18653: 18624: 18619: 18615: 18611: 18606: 18602: 18590: 18580: 18562: 18559: 18554: 18551: 18548: 18545: 18542: 18539: 18535: 18510: 18507: 18502: 18499: 18495: 18482: 18479: 18476: 18460: 18448: 18445: 18442: 18436: 18430: 18426: 18420: 18415: 18410: 18406: 18394: 18391: 18386: 18383: 18379: 18375: 18370: 18365: 18361: 18355: 18346: 18339: 18334: 18330: 18326: 18318: 18313: 18309: 18303: 18300: 18296: 18292: 18286: 18280: 18276: 18272: 18269: 18241: 18238: 18235: 18229: 18223: 18219: 18213: 18208: 18203: 18199: 18187: 18184: 18179: 18176: 18172: 18168: 18163: 18158: 18154: 18148: 18139: 18128: 18110: 18105: 18101: 18095: 18092: 18084: 18067: 18057: 18054: 18051: 18045: 18039: 18035: 18029: 18024: 18019: 18015: 18009: 18004: 17999: 17995: 17989: 17986: 17982: 17978: 17972: 17969: 17961: 17956: 17952: 17946: 17940: 17937: 17926: 17921: 17906: 17896: 17893: 17890: 17884: 17878: 17874: 17868: 17863: 17858: 17854: 17847: 17843: 17837: 17834: 17826: 17821: 17817: 17793: 17786: 17783: 17780: 17773: 17769: 17765: 17759: 17754: 17750: 17745: 17741: 17735: 17732: 17724: 17720: 17711: 17707: 17703: 17702:approximately 17699: 17681: 17677: 17668: 17654: 17648: 17643: 17639: 17633: 17630: 17626: 17622: 17619: 17613: 17610: 17584: 17581: 17554: 17549: 17545: 17539: 17536: 17532: 17528: 17522: 17516: 17509:We know that 17505: 17489: 17477: 17474: 17471: 17465: 17459: 17455: 17449: 17444: 17439: 17435: 17423: 17420: 17415: 17412: 17408: 17404: 17399: 17394: 17390: 17384: 17375: 17368: 17363: 17359: 17350: 17344: 17338: 17335: 17327: 17322: 17306: 17303: 17300: 17293: 17288: 17281: 17278: 17272: 17267: 17263: 17259: 17256: 17252: 17245: 17241: 17234: 17229: 17225: 17220: 17215: 17209: 17205: 17201: 17198: 17193: 17189: 17182: 17176: 17173: 17161: 17159: 17143: 17135: 17125: 17108: 17102: 17098: 17088: 17081: 17078: 17068: 17063: 17059: 17033: 17023: 17019: 17011: 17003: 16994: 16987: 16984: 16955: 16951: 16947: 16942: 16938: 16929: 16925: 16921: 16903: 16897: 16890: 16887: 16881: 16877: 16874: 16847: 16843: 16840: 16833: 16830: 16827: 16821: 16818: 16792: 16784: 16777: 16771: 16752: 16742: 16738: 16730: 16722: 16713: 16709: 16702: 16694: 16684: 16680: 16673: 16668: 16664: 16660: 16655: 16645: 16641: 16633: 16627: 16623: 16594: 16590: 16585: 16577: 16568: 16564: 16557: 16549: 16545: 16541: 16536: 16532: 16528: 16523: 16519: 16514: 16508: 16504: 16477: 16474: 16471: 16468: 16465: 16462: 16459: 16456: 16453: 16427: 16424: 16421: 16418: 16415: 16412: 16409: 16398: 16388: 16385: 16379: 16369: 16356: 16352: 16346: 16336: 16328: 16321: 16314: 16311: 16303: 16300: 16296: 16292: 16289: 16286: 16281: 16277: 16272: 16268: 16260: 16250: 16242: 16236: 16229: 16226: 16219: 16216: 16207: 16200: 16196: 16193: 16190: 16187: 16167: 16147: 16124: 16118: 16115: 16108:and variance 16092: 16086: 16075: 16072:is, then the 16071: 16068: 16046: 16021: 16017: 16013: 16010: 16007: 16002: 15998: 15994: 15989: 15985: 15975: 15959: 15952: 15949: 15937: 15933: 15917: 15897: 15877: 15869: 15864: 15851: 15846: 15840: 15835: 15828: 15825: 15819: 15814: 15810: 15806: 15803: 15799: 15792: 15788: 15781: 15776: 15769: 15766: 15758: 15753: 15747: 15743: 15739: 15736: 15731: 15727: 15720: 15714: 15711: 15683: 15679: 15675: 15672: 15669: 15666: 15663: 15658: 15654: 15650: 15647: 15644: 15639: 15635: 15631: 15628: 15608: 15588: 15566: 15562: 15541: 15533: 15529: 15524: 15511: 15503: 15499: 15495: 15492: 15489: 15486: 15483: 15478: 15474: 15470: 15467: 15464: 15459: 15455: 15451: 15448: 15445: 15442: 15439: 15436: 15428: 15424: 15420: 15415: 15411: 15407: 15404: 15399: 15395: 15391: 15388: 15380: 15376: 15372: 15369: 15366: 15361: 15357: 15353: 15348: 15344: 15340: 15337: 15334: 15331: 15325: 15300: 15296: 15292: 15289: 15259: 15239: 15231: 15227: 15223: 15220: 15212: 15209: 15206: 15202: 15194: 15190: 15186: 15179: 15174: 15171: 15168: 15164: 15160: 15154: 15151: 15148: 15142: 15134: 15130: 15126: 15122: 15118: 15100: 15084: 15079: 15076: 15071: 15067: 15064: 15054: 15049: 15042: 15038: 15033: 15028: 15023: 15019: 15013: 15007: 15002: 14996: 14993: 14990: 14987: 14984: 14978: 14971: 14967: 14963: 14960: 14941: 14938: 14935: 14929: 14926: 14923: 14920: 14895: 14891: 14887: 14884: 14878: 14875: 14872: 14869: 14846: 14840: 14837: 14834: 14831: 14828: 14825: 14817: 14814: 14810: 14792: 14786: 14783: 14780: 14774: 14771: 14768: 14762: 14759: 14739: 14736: 14730: 14724: 14718: 14712: 14709: 14706: 14700: 14671: 14668: 14665: 14659: 14656: 14648: 14632: 14629: 14626: 14601: 14597: 14593: 14590: 14584: 14581: 14578: 14575: 14567: 14566: 14565: 14548: 14544: 14538: 14533: 14529: 14525: 14521: 14516: 14512: 14508: 14504: 14501: 14498: 14496: 14489: 14485: 14475: 14471: 14465: 14460: 14456: 14452: 14447: 14443: 14438: 14432: 14428: 14422: 14417: 14413: 14409: 14404: 14400: 14395: 14389: 14385: 14379: 14375: 14371: 14366: 14363: 14359: 14353: 14350: 14347: 14343: 14336: 14331: 14327: 14322: 14318: 14315: 14307: 14303: 14296: 14285: 14281: 14274: 14266: 14262: 14256: 14252: 14248: 14243: 14240: 14236: 14230: 14227: 14224: 14220: 14213: 14208: 14204: 14199: 14195: 14192: 14190: 14183: 14178: 14174: 14164: 14160: 14154: 14149: 14145: 14141: 14136: 14132: 14127: 14121: 14117: 14113: 14105: 14101: 14094: 14086: 14082: 14078: 14074: 14068: 14064: 14058: 14054: 14049: 14045: 14039: 14037: 14030: 14026: 14012: 14010: 13987: 13982: 13977: 13972: 13968: 13962: 13957: 13952: 13948: 13942: 13938: 13930: 13926: 13921: 13917: 13913: 13908: 13905: 13903: 13896: 13892: 13884: 13880: 13876: 13873: 13865: 13853: 13849: 13844: 13840: 13828: 13824: 13820: 13816: 13812: 13803: 13800: 13793: 13789: 13784: 13776: 13771: 13768: 13766: 13759: 13754: 13750: 13725: 13722: 13717: 13713: 13690: 13686: 13661: 13656: 13651: 13646: 13642: 13636: 13632: 13626: 13622: 13616: 13611: 13607: 13603: 13598: 13594: 13589: 13585: 13581: 13576: 13573: 13571: 13564: 13560: 13552: 13548: 13544: 13541: 13533: 13523: 13519: 13513: 13508: 13504: 13500: 13495: 13491: 13486: 13482: 13471: 13468: 13461: 13456: 13452: 13447: 13436: 13431: 13427: 13423: 13418: 13414: 13410: 13406: 13402: 13395: 13390: 13387: 13385: 13378: 13373: 13369: 13343: 13323: 13301: 13297: 13291: 13286: 13283: 13280: 13276: 13272: 13269: 13249: 13229: 13204: 13199: 13195: 13191: 13186: 13182: 13175: 13172: 13169: 13164: 13160: 13151: 13131: 13127: 13123: 13120: 13107: 13101: 13095: 13092: 13067: 13063: 13059: 13056: 13050: 13047: 13044: 13041: 13033: 13019: 13011: 13007: 13003: 13000: 12994: 12991: 12988: 12982: 12976: 12973: 12965: 12944: 12940: 12936: 12933: 12920: 12917: 12909: 12908: 12902: 12900: 12896: 12877: 12872: 12867: 12863: 12857: 12854: 12847: 12846: 12845: 12831: 12811: 12791: 12784:The harmonic 12782: 12780: 12776: 12766: 12764: 12759: 12757: 12736: 12732: 12725: 12722: 12713: 12697: 12693: 12672: 12668: 12656: 12652: 12645: 12642: 12629: 12624: 12620: 12591: 12587: 12580: 12577: 12571: 12568: 12565: 12539: 12517: 12513: 12492: 12482: 12472: 12457: 12445: 12440: 12436: 12430: 12425: 12422: 12419: 12415: 12408: 12403: 12399: 12393: 12388: 12385: 12382: 12378: 12366: 12363: 12360: 12355: 12351: 12345: 12340: 12337: 12334: 12330: 12325: 12322: 12302: 12277: 12272: 12268: 12264: 12259: 12255: 12248: 12245: 12242: 12237: 12233: 12223: 12209: 12187: 12182: 12178: 12174: 12169: 12164: 12160: 12156: 12151: 12147: 12126: 12104: 12100: 12096: 12091: 12087: 12083: 12080: 12058: 12054: 12050: 12045: 12041: 12037: 12034: 12012: 12008: 11985: 11981: 11972: 11948: 11945: 11942: 11917: 11913: 11907: 11903: 11896: 11893: 11890: 11884: 11881: 11878: 11873: 11869: 11843: 11839: 11835: 11832: 11826: 11823: 11820: 11817: 11809: 11795: 11787: 11783: 11776: 11773: 11770: 11764: 11761: 11758: 11752: 11749: 11721: 11717: 11713: 11710: 11704: 11701: 11698: 11695: 11687: 11673: 11670: 11667: 11642: 11638: 11631: 11628: 11625: 11622: 11619: 11616: 11610: 11607: 11604: 11601: 11598: 11573: 11569: 11565: 11562: 11556: 11553: 11550: 11547: 11539: 11538: 11532: 11529: 11510: 11505: 11501: 11496: 11490: 11485: 11481: 11477: 11474: 11468: 11465: 11462: 11459: 11438: 11430: 11425: 11421: 11416: 11410: 11405: 11401: 11397: 11394: 11388: 11382: 11378: 11373: 11369: 11366: 11363: 11360: 11337: 11334: 11331: 11325: 11322: 11311: 11307: 11303: 11298: 11294: 11287: 11284: 11275: 11259: 11256: 11250: 11244: 11241: 11233: 11229: 11225: 11222: 11219: 11213: 11210: 11207: 11202: 11198: 11174: 11170: 11166: 11163: 11160: 11154: 11151: 11148: 11143: 11139: 11113: 11109: 11105: 11100: 11096: 11089: 11086: 11077: 11074: 11071: 11065: 11062: 11053: 11027: 11015: 11010: 11006: 11001: 10995: 10990: 10986: 10982: 10979: 10973: 10970: 10967: 10960: 10943: 10938: 10934: 10929: 10923: 10918: 10914: 10910: 10907: 10901: 10897: 10891: 10888: 10885: 10882: 10879: 10876: 10863: 10859: 10855: 10852: 10843: 10837: 10832: 10828: 10823: 10817: 10812: 10808: 10804: 10801: 10797: 10793: 10790: 10787: 10784: 10779: 10775: 10770: 10752: 10737: 10734: 10728: 10721: 10720: 10702: 10685: 10673: 10669: 10662: 10657: 10653: 10649: 10641: 10637: 10633: 10627: 10622: 10618: 10611: 10607: 10603: 10600: 10592: 10589: 10579: 10575: 10568: 10565: 10562: 10558: 10553: 10535: 10527: 10524: 10518: 10511: 10510: 10504: 10496: 10479: 10473: 10465: 10462: 10459: 10456: 10453: 10445: 10442: 10437: 10433: 10429: 10426: 10421: 10418: 10409: 10406: 10402: 10396: 10385: 10377: 10374: 10368: 10361: 10360: 10358: 10341: 10331: 10328: 10323: 10319: 10315: 10309: 10306: 10303: 10295: 10291: 10287: 10281: 10276: 10272: 10265: 10261: 10257: 10254: 10246: 10243: 10233: 10230: 10225: 10221: 10217: 10211: 10208: 10203: 10199: 10194: 10180: 10172: 10169: 10163: 10156: 10155: 10153: 10149: 10132: 10123: 10119: 10115: 10107: 10103: 10099: 10093: 10088: 10084: 10077: 10073: 10069: 10066: 10058: 10055: 10050: 10047: 10043: 10038: 10027: 10019: 10016: 10010: 10003: 10002: 10000: 9996: 9979: 9972: 9969: 9962: 9954: 9951: 9948: 9945: 9942: 9933: 9929: 9925: 9922: 9914: 9911: 9904: 9899: 9895: 9890: 9879: 9871: 9868: 9862: 9855: 9854: 9852: 9835: 9826: 9822: 9818: 9811: 9803: 9800: 9797: 9794: 9791: 9782: 9778: 9774: 9771: 9763: 9760: 9755: 9752: 9748: 9743: 9732: 9724: 9721: 9715: 9708: 9707: 9705: 9701: 9697: 9696: 9691: 9687: 9685: 9681: 9663: 9659: 9655: 9650: 9646: 9625: 9622: 9619: 9583: 9578: 9574: 9571: 9563: 9559: 9552: 9549: 9543: 9536: 9532: 9527: 9523: 9520: 9512: 9508: 9501: 9498: 9492: 9482: 9477: 9471: 9467: 9463: 9460: 9457: 9449: 9445: 9438: 9435: 9429: 9422: 9418: 9413: 9407: 9403: 9399: 9396: 9393: 9385: 9381: 9374: 9371: 9365: 9355: 9348: 9343: 9339: 9333: 9330: 9326: 9322: 9320: 9307: 9303: 9299: 9294: 9290: 9283: 9280: 9277: 9274: 9268: 9257: 9252: 9248: 9245: 9239: 9233: 9230: 9224: 9217: 9214: 9208: 9203: 9196: 9190: 9187: 9184: 9179: 9175: 9171: 9168: 9162: 9152: 9145: 9140: 9136: 9130: 9127: 9123: 9119: 9117: 9109: 9106: 9103: 9100: 9097: 9091: 9080: 9075: 9071: 9068: 9062: 9056: 9053: 9047: 9037: 9032: 9026: 9022: 9018: 9015: 9012: 9006: 9000: 8997: 8991: 8981: 8974: 8969: 8965: 8959: 8956: 8952: 8948: 8946: 8938: 8935: 8932: 8929: 8926: 8920: 8909: 8908: 8907: 8893: 8873: 8859: 8857: 8853: 8849: 8845: 8841: 8801: 8796: 8792: 8789: 8786: 8783: 8778: 8774: 8770: 8767: 8761: 8748: 8744: 8737: 8734: 8728: 8725: 8721: 8717: 8714: 8711: 8704: 8701: 8698: 8695: 8692: 8684: 8680: 8676: 8666: 8662: 8658: 8652: 8646: 8639: 8638: 8637: 8620: 8617: 8614: 8608: 8602: 8599: 8596: 8593: 8590: 8584: 8578: 8572: 8566: 8558: 8539: 8536: 8533: 8526: 8523: 8520: 8517: 8514: 8506: 8502: 8498: 8488: 8484: 8480: 8474: 8468: 8461: 8460: 8459: 8445: 8425: 8397: 8389: 8385: 8381: 8376: 8372: 8368: 8362: 8356: 8353: 8346: 8345: 8344: 8341: 8324: 8312: 8288: 8280: 8268: 8257: 8253: 8244: 8240: 8236: 8228: 8216: 8212: 8209: 8206: 8202: 8198: 8192: 8184: 8180: 8172: 8171: 8170: 8156: 8136: 8133: 8130: 8127: 8124: 8116: 8097: 8090: 8086: 8082: 8079: 8075: 8071: 8065: 8059: 8056: 8049: 8048: 8047: 8033: 8030: 8026: 8019: 8016: 8013: 8002: 7994: 7990: 7986: 7982: 7977: 7968: 7964: 7936: 7932: 7923: 7915: 7909: 7898: 7892: 7889: 7883: 7880: 7876: 7872: 7869: 7866: 7862: 7854: 7846: 7840: 7827: 7823: 7816: 7807: 7803: 7800: 7797: 7795: 7788: 7784: 7776: 7772: 7764: 7756: 7750: 7744: 7738: 7732: 7729: 7722: 7714: 7708: 7699: 7695: 7692: 7689: 7685: 7674: 7670: 7663: 7653: 7645: 7639: 7630: 7626: 7623: 7620: 7618: 7613: 7602: 7601: 7600: 7597: 7591: 7585: 7567: 7562: 7559: 7552: 7548: 7543: 7537: 7531: 7525: 7522: 7515: 7514: 7513: 7492: 7486: 7475: 7469: 7466: 7455:is the ratio 7439: 7433: 7430: 7423: 7400: 7395: 7392: 7385: 7381: 7376: 7366: 7362: 7355: 7352: 7346: 7343: 7339: 7335: 7330: 7327: 7320: 7316: 7311: 7302: 7296: 7290: 7282: 7276: 7273: 7268: 7266: 7258: 7252: 7249: 7242: 7236: 7233: 7226: 7222: 7217: 7206: 7202: 7198: 7195: 7192: 7188: 7184: 7178: 7175: 7168: 7164: 7159: 7150: 7139: 7133: 7124: 7119: 7111: 7105: 7099: 7091: 7087: 7080: 7074: 7072: 7064: 7058: 7055: 7048: 7041: 7037: 7033: 7030: 7027: 7024: 7020: 7016: 7014: 7004: 7000: 6993: 6983: 6976: 6972: 6965: 6962: 6956: 6953: 6949: 6945: 6943: 6935: 6929: 6915: 6914: 6913: 6910: 6890: 6883: 6879: 6873: 6869: 6863: 6860: 6855: 6852: 6849: 6845: 6841: 6833: 6829: 6822: 6812: 6811: 6810: 6807: 6802: 6797: 6791: 6780: 6778: 6774: 6770: 6750: 6746: 6740: 6737: 6732: 6728: 6704: 6696: 6690: 6687: 6682: 6676: 6670: 6667: 6664: 6655: 6651: 6646: 6640: 6635: 6631: 6627: 6620: 6616: 6610: 6607: 6602: 6599: 6595: 6591: 6585: 6579: 6569: 6568: 6567: 6565: 6561: 6556: 6554: 6538: 6530: 6514: 6511: 6506: 6502: 6498: 6492: 6486: 6483: 6475: 6455: 6451: 6446: 6442: 6436: 6430: 6427: 6418: 6402: 6398: 6394: 6389: 6385: 6381: 6375: 6369: 6366: 6358: 6340: 6336: 6332: 6327: 6323: 6319: 6313: 6307: 6304: 6296: 6286: 6284: 6274: 6272: 6246: 6243: 6240: 6220: 6217: 6212: 6208: 6204: 6201: 6181: 6178: 6175: 6172: 6169: 6146: 6143: 6140: 6137: 6134: 6128: 6108: 6105: 6102: 6099: 6096: 6093: 6090: 6070: 6061: 6052: 6038: 6030: 6014: 5983: 5979: 5973: 5969: 5965: 5962: 5959: 5953: 5950: 5947: 5941: 5932: 5928: 5924: 5914: 5910: 5904: 5900: 5896: 5893: 5890: 5884: 5881: 5878: 5870: 5866: 5860: 5856: 5852: 5849: 5846: 5838: 5834: 5827: 5823: 5819: 5816: 5810: 5804: 5798: 5791: 5790: 5789: 5775: 5752: 5746: 5737: 5735: 5714: 5710: 5704: 5700: 5694: 5690: 5686: 5683: 5680: 5676: 5669: 5666: 5659: 5651: 5648: 5632: 5629: 5626: 5622: 5614: 5613: 5612: 5610: 5609:formal series 5606: 5601: 5595: 5573: 5570: 5567: 5563: 5556: 5546: 5541: 5527: 5502: 5499: 5495: 5488: 5461: 5452: 5448: 5444: 5441: 5438: 5432: 5426: 5420: 5417: 5410: 5407: 5382: 5378: 5372: 5368: 5362: 5358: 5354: 5351: 5348: 5344: 5340: 5332: 5328: 5321: 5311: 5310: 5309: 5301: 5299: 5280: 5274: 5271: 5264: 5261: 5252: 5238: 5235: 5227: 5222: 5219: 5206: 5203: 5198: 5193: 5189: 5185: 5180: 5176: 5171: 5167: 5162: 5159: 5143: 5140: 5133: 5132: 5131: 5129: 5110: 5103: 5100: 5090: 5087: 5082: 5077: 5073: 5068: 5064: 5059: 5043: 5033: 5032: 5031: 4968: 4964: 4957: 4954: 4951: 4946: 4942: 4933: 4908: 4890: 4871: 4869: 4843: 4834: 4829: 4824: 4821: 4818: 4815: 4812: 4806: 4802: 4798: 4795: 4790: 4787: 4782: 4778: 4773: 4765: 4760: 4755: 4752: 4749: 4746: 4743: 4737: 4733: 4730: 4727: 4724: 4720: 4714: 4711: 4702: 4701: 4700: 4697: 4677: 4671: 4668: 4659: 4599: 4594: 4590: 4587: 4581: 4578: 4575: 4566: 4559: 4553: 4545: 4541: 4533: 4532: 4531: 4529: 4501: 4494: 4485: 4481: 4474: 4464: 4456: 4453: 4450: 4447: 4444: 4432: 4428: 4424: 4421: 4407: 4401: 4396: 4390: 4383: 4378: 4376: 4362: 4356: 4349: 4343: 4338: 4331: 4328: 4325: 4322: 4319: 4310: 4304: 4300: 4298: 4289: 4284: 4277: 4274: 4271: 4268: 4265: 4256: 4249: 4230: 4225: 4221: 4218: 4215: 4212: 4209: 4203: 4197: 4193: 4191: 4182: 4177: 4170: 4167: 4164: 4161: 4158: 4149: 4135: 4117: 4115: 4097: 4094: 4091: 4088: 4085: 4082: 4079: 4042: 4024: 4022: 4004: 4001: 3998: 3961: 3943: 3941: 3933: 3925: 3921: 3909: 3908: 3907: 3905: 3880: 3877: 3874: 3835: 3778: 3774: 3770: 3767: 3754: 3748: 3742: 3739: 3732: 3731: 3730: 3717: 3709: 3705: 3694:and variance 3681: 3658: 3631: 3622: 3618: 3614: 3611: 3604: 3600: 3597: 3594: 3591: 3565: 3548: 3529: 3525: 3513: 3512: 3511: 3488: 3484: 3455: 3451: 3447: 3442: 3438: 3409: 3405: 3401: 3396: 3392: 3379: 3366: 3359: 3351: 3346: 3342: 3332: 3327: 3323: 3313: 3310: 3306: 3302: 3299: 3296: 3291: 3287: 3259: 3243: 3238: 3234: 3230: 3225: 3220: 3216: 3204: 3199: 3195: 3189: 3185: 3182: 3179: 3176: 3153: 3145: 3140: 3136: 3125:and variance 3110: 3106: 3096: 3082: 3079: 3076: 3073: 3070: 3050: 3042: 3038: 3009: 3005: 2994:Likewise, if 2981: 2975: 2972: 2969: 2966: 2963: 2934: 2928: 2923: 2919: 2889: 2883: 2878: 2874: 2841: 2837: 2833: 2830: 2804: 2801: 2798: 2795: 2792: 2789: 2777: 2773: 2756: 2745: 2741: 2737: 2733: 2729: 2713: 2693: 2668: 2665: 2662: 2659: 2655: 2651: 2648: 2641: 2640: 2639: 2625: 2622: 2619: 2599: 2579: 2571: 2552: 2530: 2526: 2516: 2512: 2508: 2504: 2501: 2497: 2492: 2490: 2486: 2482: 2478: 2474: 2470: 2465: 2463: 2459: 2455: 2447: 2443: 2433: 2422: 2418: 2408: 2404: 2400: 2396: 2392: 2388: 2384: 2380: 2353: 2350: 2344: 2336: 2333: 2319: 2316: 2307: 2304: 2301: 2297: 2288: 2283: 2279: 2273: 2270: 2266: 2262: 2250: 2235: 2227: 2220: 2217: 2214: 2211: 2205: 2200: 2197: 2193: 2189: 2175: 2166: 2160: 2156: 2153: 2150: 2147: 2134: 2129: 2125: 2119: 2116: 2112: 2106: 2103: 2088: 2084: 2063: 2056: 2053: 2042: 2034: 2028: 2004: 1998: 1995: 1985: 1981: 1978: 1973: 1970: 1946: 1939: 1925: 1922: 1911: 1903: 1897: 1873: 1867: 1864: 1842: 1836: 1823: 1819: 1816: 1813: 1810: 1798: 1794: 1774: 1760: 1756: 1748: 1741: 1734: 1721: 1717: 1709: 1701: 1687: 1683: 1659: 1655: 1649: 1645: 1639: 1635: 1631: 1628: 1625: 1618: 1611: 1608: 1598: 1590: 1584: 1565: 1562: 1559: 1555: 1541: 1537: 1529: 1525: 1506: 1494: 1491: 1485: 1482: 1478: 1471: 1460: 1457: 1448: 1444: 1439: 1435: 1422: 1418: 1400: 1397: 1393: 1387: 1383: 1379: 1375: 1371: 1368: 1362: 1359: 1355: 1349: 1345: 1338: 1334: 1330: 1327: 1321: 1318: 1314: 1308: 1304: 1297: 1293: 1289: 1286: 1280: 1268: 1264: 1246: 1243: 1235: 1231: 1224: 1221: 1215: 1208: 1205: 1201: 1196: 1192: 1188: 1184: 1181: 1174: 1161: 1157: 1138: 1132: 1128: 1124: 1121: 1115: 1111: 1107: 1104: 1097: 1090: 1087: 1079: 1075: 1068: 1065: 1058: 1045: 1041: 1022: 1013: 1009: 1005: 1002: 995: 991: 988: 976: 972: 948: 939: 936: 924: 920: 901: 892: 887: 883: 877: 874: 867: 863: 860: 848: 844: 823: 815: 812: 804: 801: 798: 792: 789: 786: 761: 754: 751: 748: 745: 739: 734: 731: 727: 719: 715: 711: 706: 703: 699: 695: 692: 680: 676: 660: 655: 651: 648: 642: 636: 633: 627: 620: 616: 611: 600: 595: 587: 584: 581: 578: 575: 566: 562: 559: 556: 553: 549: 543: 536: 518: 514: 498: 489: 485: 481: 475: 470: 463: 460: 457: 454: 451: 447: 440: 436: 432: 429: 412: 409: 404: 401: 394: 380: 376: 349: 346: 343: 334: 331: 319: 315: 311: 292: 289: 286: 274: 246: 243: 237: 228: 225: 213: 209: 190: 181: 177: 172: 169: 162: 158: 155: 141: 122: 119: 116: 103: 95: 79: 56: 42: 34: 19: 28031: 28019: 27985:Multivariate 27984: 27972: 27960: 27955:Wrapped Lévy 27915: 27863:Matrix gamma 27856: 27836: 27824:Normal-gamma 27817: 27783:Continuous: 27782: 27753: 27698:Tukey lambda 27685: 27677: 27672:-exponential 27669: 27661: 27652: 27643: 27634: 27628:-exponential 27625: 27569: 27536: 27503: 27465: 27452: 27379:Poly-Weibull 27328: 27324:Log-logistic 27284: 27283:Hotelling's 27215: 27057:Logit-normal 26931:Gauss–Kuzmin 26926:Flory–Schulz 26807:with finite 26727: 26723: 26720:Donal, Wales 26702: 26698: 26679: 26675: 26663: 26630: 26612: 26603: 26582: 26558:(1): 59–70. 26555: 26551: 26541: 26498: 26494: 26488: 26455: 26451: 26441: 26433:the original 26428: 26419: 26400: 26394: 26369: 26365: 26355: 26330: 26326: 26316: 26271: 26267: 26257: 26222: 26218: 26212: 26204: 26199: 26179: 26172: 26147: 26143: 26137: 26113:. Springer. 26110: 26104: 26087: 26076: 26052: 26038: 26008:(2): 49–73. 26005: 26001: 25991: 25948: 25944: 25934: 25899: 25895: 25885: 25848: 25844: 25834: 25791: 25787: 25777: 25734: 25731:Cell Reports 25730: 25720: 25711: 25705: 25672: 25668: 25658: 25633: 25629: 25623: 25588: 25584: 25574: 25541: 25537: 25531: 25523: 25518: 25491: 25485: 25473:. Retrieved 25469: 25460: 25450: 25443: 25434: 25380: 25376: 25336: 25332: 25303:(1): 40–59. 25300: 25296: 25290: 25257: 25253: 25247: 25186: 25182: 25172: 25147: 25143: 25137: 25113:(56): 1–14. 25110: 25106: 25082: 25073: 25064: 25052: 25015: 25011: 25005: 24972: 24966: 24957: 24940: 24936: 24930: 24921:1601.01763v1 24909: 24872: 24866: 24849: 24845: 24839: 24822: 24818: 24805: 24786: 24782: 24744: 24740: 24734: 24717: 24713: 24707: 24690: 24686: 24680: 24668: 24659: 24634: 24630: 24624: 24599: 24595: 24589: 24570: 24566: 24556: 24547: 24500: 24496: 24486: 24474:. Retrieved 24465: 24400: 24394: 24359: 24355: 24345: 24340:, Paper PO08 24337: 24321: 24284: 24280: 24248: 24244: 24234: 24220:(4): 908–9. 24217: 24213: 24159: 24155: 24125: 24106: 24102: 24089: 24080:1721.1/48703 24065:(1): 37–55. 24062: 24058: 24045: 24020: 24016: 24010: 23993: 23989: 23952: 23945: 23917: 23911: 23891: 23884: 23864: 23857: 23846:. Retrieved 23839:the original 23816: 23812: 23799: 23769: 23743:. Retrieved 23734: 23711:. Retrieved 23707: 23679:. Retrieved 23641: 23635: 23622: 22981: 22295:delta method 22289: 22280: 22267: 22199:ball milling 22121: 21716: 21712: 21704:Gibrat's law 21696: 21648: 21595: 21538: 21533: 21044: 21029: 21025:As desired. 21024: 20223: 19842: 19701: 19696: 19213: 19210: 18964: 18961: 18950: 18752: 18749: 18586: 18485: 18480: 18478:As desired. 18477: 17922: 17669: 17508: 17323: 17162: 17131: 16927: 16919: 16780: 16775: 16394: 16381: 16069: 15976: 15935: 15931: 15867: 15865: 15531: 15527: 15525: 15135:. Note that 15124: 15120: 15114: 14646: 14013: 14006: 12892: 12804:, geometric 12783: 12775:Lorenz curve 12772: 12763:Gibrat's law 12760: 12714: 12484: 12481:Gibrat's law 12224: 11968: 11530: 11276: 11054: 11050: 10703:LogNormal7(μ 9611: 8865: 8817: 8554: 8417: 8342: 8303: 8112: 7998: 7962: 7955: 7595: 7589: 7586: 7582: 7419: 6908: 6905: 6809:is given by 6805: 6795: 6789: 6786: 6719: 6557: 6528: 6419: 6292: 6280: 6267: 6006: 5738: 5731: 5599: 5593: 5542: 5399: 5307: 5295: 5125: 4877: 4859: 4698: 4615: 4525: 3798: 3554: 3508: 3507: 3380: 3097: 2862: 2743: 2685: 2541: 2524: 2511:Gibrat's law 2493: 2489:Cobb–Douglas 2472: 2468: 2466: 2445: 2441: 2420: 2416: 2391:distribution 2390: 2386: 2382: 2376: 2091: 28069:Exponential 27918:directional 27907:Directional 27794:Generalized 27765:Multinomial 27720:continuous- 27660:Kaniadakis 27651:Kaniadakis 27642:Kaniadakis 27633:Kaniadakis 27624:Kaniadakis 27576:Tracy–Widom 27553:Skew normal 27535:Noncentral 27319:Log-Laplace 27297:Generalized 27278:Half-normal 27244:Generalized 27208:Logarithmic 27193:Exponential 27147:Chi-squared 27087:U-quadratic 27052:Kumaraswamy 26994:Continuous 26941:Logarithmic 26836:Categorical 26425:"Shadowing" 26274:(1): 2811. 25794:: 156–193. 25260:: 118–130. 22181:reliability 22135:heavy tails 21950:, then the 19828:, which is 17925:also normal 17700:, which is 15866:For finite 15131:as for the 12712:is finite. 12685:, assuming 12532:shows, for 11971:independent 6566:. In fact, 6283:Matlab code 6271:Matlab code 2533:Definitions 2500:independent 2454:engineering 28147:Categories 28064:Elliptical 28020:Degenerate 28006:Degenerate 27754:Discrete: 27713:univariate 27568:Student's 27523:Asymmetric 27502:Johnson's 27430:supported 27374:Phase-type 27329:Log-normal 27314:Log-Cauchy 27304:Kolmogorov 27222:Noncentral 27152:Noncentral 27132:Beta prime 27082:Triangular 27077:Reciprocal 27047:Irwin–Hall 26996:univariate 26976:Yule–Simon 26858:Rademacher 26800:univariate 26676:BioScience 26657:0644.62014 26594:1902.03853 26501:(28): 28. 26281:1703.06645 26232:1510.08877 26150:(3): 637. 25974:2164/10561 25494:. London. 25339:(5): 341. 25333:BioScience 24968:Biometrika 24882:1705.03196 24567:BioScience 24471:"ProbOnto" 24362:(8): 1–3. 24245:BioScience 24214:Biometrics 24169:2012.14331 23848:2011-06-02 23745:2020-09-13 23713:2020-09-13 23681:2023-02-27 23651:1811.11301 23615:References 22210:MIME types 22174:Technology 22160:City sizes 21649:The value 21040:consistent 19838:parameters 18957:odds-ratio 17710:parameters 15097:This is a 12777:(see also 12479:See also: 11810:Power: If 8113:Since the 6055:Properties 5611:diverges: 2572:, and let 2383:log-normal 212:Parameters 27789:Dirichlet 27770:Dirichlet 27680:-Gaussian 27655:-Logistic 27492:Holtsmark 27464:Gaussian 27451:Fisher's 27434:real line 26936:Geometric 26916:Delaporte 26821:Bernoulli 26798:Discrete 26574:0163-5999 26508:1111.6849 26480:2058-1106 26164:154552078 26099:, EconWPA 26030:0016-7037 25983:1367-2630 25808:1873-5118 25751:2211-1247 25689:1054-3406 25607:1367-4803 25510:476909537 25397:1471-003X 25355:0006-3568 25282:117598888 25221:1674-7348 25196:1304.5603 24761:0346-1238 24447:ignored ( 24437:cite book 24162:(10): 1. 23972:780289503 23821:CiteSeerX 23668:254231768 23584:− 23535:− 23404:− 23358:^ 23355:μ 23348:− 23336:^ 23333:μ 23307:^ 23290:− 23259:σ 23241:− 23210:σ 23178:σ 23146:σ 23102:^ 23072:^ 23050:^ 22904:^ 22863:∗ 22833:− 22824:∗ 22762:^ 22720:^ 22682:∗ 22670:^ 22640:^ 22630:− 22621:± 22606:^ 22528:∗ 22498:− 22452:˙ 22449:∼ 22431:^ 22372:− 22331:˙ 22328:∼ 22313:^ 22206:file size 22192:shadowing 22122:logarithm 22088:Φ 22080:, where 22065:− 22051:σ 22042:Φ 21980:σ 21971:⁡ 21938:σ 21917:economics 21865:hydrology 21858:Hydrology 21824:Chemistry 21657:σ 21634:σ 21614:… 21567:σ 21547:σ 21501:− 21470:σ 21452:− 21421:σ 21389:σ 21357:σ 21319:σ 21315:− 21301:σ 21272:μ 21268:− 21259:μ 21220:− 21174:^ 21171:μ 21164:− 21152:^ 21149:μ 21098:^ 21068:^ 20993:− 20944:− 20834:α 20829:− 20818:± 20787:− 20747:^ 20744:μ 20737:− 20725:^ 20722:μ 20674:σ 20659:μ 20632:σ 20617:μ 20518:− 20469:− 20359:α 20354:− 20343:± 20312:− 20272:^ 20269:μ 20262:− 20250:^ 20247:μ 20196:− 20165:σ 20147:− 20116:σ 20084:σ 20052:σ 20029:σ 20025:− 20011:σ 19982:μ 19978:− 19969:μ 19954:∼ 19923:− 19883:^ 19880:μ 19873:− 19861:^ 19858:μ 19739:^ 19736:μ 19729:− 19717:^ 19714:μ 19665:− 19616:− 19506:α 19501:− 19490:± 19459:− 19419:^ 19416:μ 19409:− 19397:^ 19394:μ 19346:σ 19331:μ 19304:σ 19289:μ 19181:σ 19177:− 19163:σ 19134:μ 19130:− 19121:μ 19086:σ 19071:μ 19044:σ 19029:μ 18872:α 18867:− 18856:± 18844:^ 18841:μ 18834:− 18822:^ 18819:μ 18788:μ 18784:− 18775:μ 18667:σ 18654:μ 18616:σ 18603:μ 18560:α 18555:− 18543:− 18508:α 18503:− 18446:− 18392:α 18387:− 18376:± 18350:^ 18347:μ 18310:σ 18301:μ 18239:− 18185:α 18180:− 18169:± 18143:^ 18140:μ 18102:σ 18093:μ 18055:− 18036:σ 18016:σ 17996:σ 17987:μ 17973:˙ 17970:∼ 17941:^ 17938:μ 17894:− 17875:σ 17855:σ 17838:˙ 17835:∼ 17807:. Hence, 17784:− 17770:σ 17751:σ 17736:˙ 17733:∼ 17640:σ 17631:μ 17620:∼ 17614:^ 17611:μ 17585:^ 17582:μ 17546:σ 17537:μ 17475:− 17421:α 17416:− 17405:± 17379:^ 17376:μ 17304:− 17282:^ 17279:μ 17273:− 17260:⁡ 17242:∑ 17202:⁡ 17190:∑ 17177:^ 17174:μ 17158:bootstrap 17144:μ 17089:∗ 17082:^ 17079:σ 17064:∗ 17024:∗ 17004:× 16995:∗ 16988:^ 16985:μ 16956:μ 16943:∗ 16939:μ 16891:^ 16888:σ 16848:^ 16834:⋅ 16828:± 16822:^ 16819:μ 16793:μ 16743:∗ 16739:σ 16723:× 16714:∗ 16710:μ 16685:∗ 16681:σ 16674:⋅ 16669:∗ 16665:μ 16646:∗ 16642:σ 16628:∗ 16624:μ 16595:∗ 16591:σ 16578:× 16569:∗ 16565:μ 16550:∗ 16546:σ 16542:⋅ 16537:∗ 16533:μ 16524:∗ 16520:σ 16509:∗ 16505:μ 16478:σ 16469:μ 16463:σ 16457:− 16454:μ 16428:σ 16422:μ 16416:σ 16413:− 16410:μ 16340:¯ 16315:^ 16312:σ 16293:⁡ 16278:σ 16254:¯ 16230:^ 16227:σ 16211:¯ 16197:⁡ 16188:μ 16168:σ 16148:μ 16119:⁡ 16087:⁡ 16050:¯ 16011:… 15953:^ 15950:σ 15918:σ 15898:σ 15878:μ 15829:^ 15826:μ 15820:− 15807:⁡ 15789:∑ 15770:^ 15767:σ 15740:⁡ 15728:∑ 15715:^ 15712:μ 15676:⁡ 15667:… 15651:⁡ 15632:⁡ 15609:σ 15589:μ 15563:ℓ 15542:ℓ 15496:⁡ 15487:… 15471:⁡ 15452:⁡ 15446:∣ 15443:σ 15437:μ 15425:ℓ 15408:⁡ 15396:∑ 15392:− 15370:… 15341:∣ 15338:σ 15332:μ 15326:ℓ 15297:σ 15290:μ 15260:φ 15224:⁡ 15213:σ 15207:μ 15203:φ 15165:∏ 15155:σ 15149:μ 15077:− 15050:σ 15039:π 15024:μ 14997:σ 14991:μ 14942:σ 14936:μ 14930:⁡ 14924:∼ 14892:σ 14885:μ 14879:⁡ 14876:Lognormal 14873:∼ 14841:⁡ 14835:∼ 14829:∣ 14787:⁡ 14763:⁡ 14725:⁡ 14701:⁡ 14675:∞ 14660:∈ 14598:σ 14591:μ 14585:⁡ 14582:Lognormal 14579:∼ 14530:σ 14526:− 14505:⁡ 14486:μ 14457:σ 14444:μ 14414:σ 14401:μ 14386:σ 14376:σ 14372:⁡ 14344:∑ 14297:⁡ 14275:⁡ 14263:σ 14253:σ 14249:⁡ 14221:∑ 14175:σ 14146:σ 14133:μ 14118:∑ 14095:⁡ 14083:∑ 14055:∑ 14046:⁡ 13969:σ 13963:− 13949:σ 13927:μ 13918:∑ 13893:μ 13850:μ 13841:∑ 13825:μ 13813:∑ 13801:− 13790:σ 13751:σ 13726:σ 13714:σ 13643:σ 13637:− 13608:σ 13595:μ 13586:∑ 13561:μ 13505:σ 13492:μ 13483:∑ 13469:− 13453:σ 13428:σ 13415:μ 13403:∑ 13370:σ 13277:∑ 13250:μ 13230:σ 13196:σ 13183:μ 13176:⁡ 13173:Lognormal 13170:∼ 13128:σ 13121:μ 13108:∼ 13096:⁡ 13064:σ 13057:μ 13051:⁡ 13048:Lognormal 13045:∼ 13008:σ 13001:μ 12995:⁡ 12992:Lognormal 12989:∼ 12977:⁡ 12941:σ 12934:μ 12921:∼ 12726:⁡ 12694:σ 12646:⁡ 12621:σ 12581:⁡ 12566:μ 12546:∞ 12543:→ 12437:σ 12416:∑ 12400:μ 12379:∑ 12367:⁡ 12364:Lognormal 12361:∼ 12331:∏ 12269:σ 12256:μ 12249:⁡ 12246:Lognormal 12243:∼ 12179:σ 12161:σ 12148:σ 12127:σ 12101:μ 12097:− 12088:μ 12081:μ 12055:μ 12042:μ 12035:μ 11946:≠ 11914:σ 11894:μ 11885:⁡ 11882:Lognormal 11879:∼ 11840:σ 11833:μ 11827:⁡ 11824:Lognormal 11821:∼ 11784:σ 11774:μ 11771:− 11765:⁡ 11762:Lognormal 11759:∼ 11718:σ 11711:μ 11705:⁡ 11702:Lognormal 11699:∼ 11639:σ 11626:⁡ 11617:μ 11611:⁡ 11608:Lognormal 11605:∼ 11570:σ 11563:μ 11557:⁡ 11554:Lognormal 11551:∼ 11502:μ 11482:σ 11469:⁡ 11422:μ 11402:σ 11379:μ 11370:⁡ 11361:μ 11332:μ 11326:⁡ 11320:→ 11308:σ 11295:μ 11288:⁡ 11257:− 11245:⁡ 11220:μ 11214:⁡ 11199:σ 11161:μ 11155:⁡ 11140:μ 11110:σ 11097:μ 11090:⁡ 11084:→ 11072:μ 11066:⁡ 11052:results. 11007:μ 10987:σ 10974:⁡ 10935:μ 10915:σ 10898:μ 10892:⁡ 10886:− 10880:⁡ 10864:− 10856:⁡ 10829:μ 10809:σ 10794:⁡ 10788:π 10758:σ 10743:μ 10670:σ 10663:⁡ 10628:⁡ 10612:− 10604:⁡ 10593:π 10576:σ 10569:⁡ 10541:σ 10466:μ 10463:− 10457:⁡ 10443:τ 10438:− 10430:⁡ 10410:π 10403:τ 10390:τ 10382:μ 10310:⁡ 10282:⁡ 10266:− 10258:⁡ 10247:π 10212:⁡ 10120:σ 10094:⁡ 10078:− 10070:⁡ 10059:π 10051:σ 10032:σ 9955:μ 9952:− 9946:⁡ 9934:− 9926:⁡ 9915:π 9876:μ 9823:σ 9804:μ 9801:− 9795:⁡ 9783:− 9775:⁡ 9764:π 9756:σ 9737:σ 9729:μ 9702:, μ, and 9664:∗ 9660:σ 9651:∗ 9647:μ 9626:σ 9620:μ 9579:σ 9575:μ 9572:− 9553:⁡ 9540:Φ 9537:− 9528:σ 9524:μ 9521:− 9502:⁡ 9489:Φ 9478:σ 9468:σ 9464:− 9461:μ 9458:− 9439:⁡ 9426:Φ 9423:− 9414:σ 9404:σ 9400:− 9397:μ 9394:− 9375:⁡ 9362:Φ 9356:⋅ 9340:σ 9331:μ 9284:∈ 9278:∣ 9253:σ 9249:μ 9246:− 9234:⁡ 9221:Φ 9218:− 9204:σ 9191:⁡ 9185:− 9176:σ 9169:μ 9159:Φ 9153:⋅ 9137:σ 9128:μ 9107:⩾ 9101:∣ 9076:σ 9072:μ 9069:− 9057:⁡ 9044:Φ 9033:σ 9023:σ 9019:− 9016:μ 9013:− 9001:⁡ 8988:Φ 8982:⋅ 8966:σ 8957:μ 8930:∣ 8852:economics 8848:insurance 8844:Talk page 8826:Φ 8797:σ 8790:⁡ 8784:− 8775:σ 8768:μ 8757:Φ 8745:σ 8726:μ 8696:∣ 8672:∞ 8663:∫ 8594:∣ 8585:⁡ 8518:∣ 8494:∞ 8485:∫ 8390:∗ 8386:μ 8377:μ 8357:⁡ 8325:α 8317:Φ 8281:α 8273:Φ 8258:∗ 8254:σ 8245:∗ 8241:μ 8229:α 8221:Φ 8213:σ 8207:μ 8193:α 8134:⁡ 8117:variable 8087:σ 8083:− 8080:μ 8060:⁡ 8017:⁡ 7910:⁡ 7893:⁡ 7873:⁡ 7841:⁡ 7817:⁡ 7804:⁡ 7785:σ 7751:⁡ 7733:⁡ 7709:⁡ 7696:⁡ 7664:⁡ 7640:⁡ 7627:⁡ 7614:μ 7560:− 7549:σ 7526:⁡ 7487:⁡ 7470:⁡ 7434:⁡ 7393:− 7382:σ 7363:σ 7344:μ 7328:− 7317:σ 7297:⁡ 7277:⁡ 7253:⁡ 7234:− 7223:σ 7203:σ 7196:μ 7176:− 7165:σ 7134:⁡ 7106:⁡ 7100:− 7081:⁡ 7059:⁡ 7038:σ 7028:μ 6994:⁡ 6973:σ 6954:μ 6930:⁡ 6880:σ 6853:μ 6823:⁡ 6747:σ 6733:− 6691:⁡ 6683:⋅ 6671:⁡ 6652:σ 6641:⋅ 6636:μ 6617:σ 6600:μ 6580:⁡ 6529:analogous 6512:− 6507:σ 6487:⁡ 6452:σ 6431:⁡ 6403:∗ 6399:σ 6390:σ 6370:⁡ 6341:∗ 6337:μ 6328:μ 6308:⁡ 6244:⁡ 6205:⁡ 6179:⁡ 6138:⁡ 6103:σ 6091:μ 6039:φ 5984:μ 5970:σ 5960:− 5929:σ 5915:μ 5901:σ 5891:− 5871:μ 5857:σ 5847:− 5828:− 5820:⁡ 5811:≈ 5799:φ 5747:φ 5701:σ 5684:μ 5638:∞ 5623:∑ 5557:⁡ 5489:⁡ 5462:σ 5449:σ 5439:μ 5433:− 5421:⁡ 5369:σ 5352:μ 5322:⁡ 5272:− 5258:Σ 5232:Σ 5216:Σ 5190:μ 5177:μ 5144:⁡ 5097:Σ 5074:μ 5044:⁡ 4958:⁡ 4914:Σ 4905:μ 4891:∼ 4830:σ 4825:μ 4822:− 4816:⁡ 4807:− 4799:⁡ 4761:σ 4756:μ 4753:− 4747:⁡ 4734:⁡ 4660:⁡ 4627:Φ 4595:σ 4591:μ 4588:− 4579:⁡ 4563:Φ 4482:σ 4457:μ 4454:− 4448:⁡ 4433:− 4425:⁡ 4408:π 4397:σ 4357:σ 4339:σ 4332:μ 4329:− 4323:⁡ 4305:φ 4285:σ 4278:μ 4275:− 4269:⁡ 4226:σ 4222:μ 4219:− 4213:⁡ 4198:φ 4178:σ 4171:μ 4168:− 4162:⁡ 4143:Φ 4095:⁡ 4089:≤ 4083:⁡ 4002:≤ 3836:φ 3810:Φ 3775:σ 3768:μ 3755:∼ 3743:⁡ 3706:σ 3682:μ 3619:σ 3612:μ 3601:⁡ 3598:Lognormal 3595:∼ 3530:∗ 3526:σ 3489:∗ 3485:μ 3456:σ 3443:∗ 3439:σ 3410:μ 3397:∗ 3393:μ 3343:μ 3324:σ 3303:⁡ 3288:σ 3235:σ 3217:μ 3196:μ 3186:⁡ 3177:μ 3166:one uses 3137:σ 3107:μ 3080:≠ 2973:≠ 2929:⁡ 2884:⁡ 2834:∼ 2802:σ 2796:μ 2740:logarithm 2714:σ 2694:μ 2666:σ 2660:μ 2620:σ 2600:σ 2580:μ 2481:McAlister 2462:economics 2403:logarithm 2387:lognormal 2354:σ 2351:− 2334:− 2330:Φ 2323:Φ 2320:− 2305:− 2280:σ 2271:μ 2218:− 2206:⁡ 2198:− 2167:σ 2157:⁡ 2126:σ 2117:μ 2029:⁡ 1999:⁡ 1982:⁡ 1971:σ 1898:⁡ 1868:⁡ 1837:⁡ 1820:⁡ 1811:μ 1757:σ 1718:σ 1646:σ 1629:μ 1571:∞ 1556:∑ 1483:μ 1472:σ 1461:π 1445:⁡ 1398:− 1384:σ 1372:⁡ 1346:σ 1331:⁡ 1305:σ 1290:⁡ 1244:− 1232:σ 1225:⁡ 1193:σ 1185:⁡ 1129:σ 1122:μ 1108:⁡ 1088:− 1076:σ 1069:⁡ 1010:σ 1006:− 1003:μ 992:⁡ 949:μ 940:⁡ 884:σ 875:μ 864:⁡ 813:− 809:Φ 805:σ 799:μ 793:⁡ 752:− 740:⁡ 732:− 716:σ 704:μ 696:⁡ 656:σ 652:μ 649:− 637:⁡ 624:Φ 596:σ 588:μ 585:− 579:⁡ 563:⁡ 486:σ 464:μ 461:− 455:⁡ 441:− 433:⁡ 413:π 405:σ 353:∞ 335:∈ 287:σ 250:∞ 241:∞ 238:− 229:∈ 226:μ 178:σ 170:μ 159:⁡ 156:Lognormal 117:μ 80:σ 57:μ 18:Lognormal 28122:Category 28054:Circular 28047:Families 28032:Singular 28011:singular 27775:Negative 27722:discrete 27688:-Weibull 27646:-Weibull 27530:Logistic 27414:Discrete 27384:Rayleigh 27364:Nakagami 27287:-squared 27261:Gompertz 27110:interval 26846:Negative 26831:Binomial 26533:17404692 26308:29434232 25926:37409647 25917:10438924 25877:29071065 25851:: 1222. 25826:23123501 25769:23994479 25615:23129296 25566:46599163 25558:16345017 25475:14 April 25415:24569488 25239:32288765 24997:18132090 24789:: 1–28. 24651:11275334 24616:22640817 24527:27153608 24386:24728329 24330:Archived 24313:28111557 24287:: 2013. 24196:34468706 23900:Archived 23873:Archived 23672:Archived 22242:See also 22143:fat tail 22025:, since 21798:and the 21783:Several 18589:A/B test 17708:), with 17571:. Also, 16866:, where 14838:Rayleigh 12139:, where 9680:ProbOnto 8027:′ 7993:skewness 6472:, and a 5605:analytic 5126:and its 2772:itself. 2498:of many 2475:, after 2458:medicine 1160:Skewness 1044:Variance 679:Quantile 273:location 143:Notation 28132:Commons 28104:Wrapped 28099:Tweedie 28094:Pearson 28089:Mixture 27996:Bingham 27895:Complex 27885:Inverse 27875:Wishart 27868:Inverse 27855:Matrix 27829:Inverse 27745:(joint) 27664:-Erlang 27518:Laplace 27409:Weibull 27266:Shifted 27249:Inverse 27234:Fréchet 27157:Inverse 27092:Uniform 27012:Arcsine 26971:Skellam 26966:Poisson 26889:support 26863:Soliton 26816:Benford 26809:support 26649:0939191 26513:Bibcode 26460:Bibcode 26386:8614756 26347:3592829 26299:5809396 26249:8338485 26095:(2005) 26058:175–224 26010:Bibcode 25953:Bibcode 25868:5639933 25817:5985929 25760:3804159 25697:9056596 25406:4051294 25309:2729692 25262:Bibcode 25230:7111546 25201:Bibcode 25152:Bibcode 25115:Bibcode 25107:Entropy 24989:2332539 24687:Ecology 24518:5013898 24428:1059583 24377:4112421 24304:5216879 24226:2530139 24187:8419883 24025:Bibcode 23792:1299979 23737:. U.S. 22114:finance 22021:is the 21876:CumFreq 14913:, then 12966:, then 11969:If two 8838:is the 6027:is the 4934:, then 4866:is the 3506:is the 2496:product 1421:Entropy 318:Support 28038:Cantor 27880:Normal 27711:Mixed 27637:-Gamma 27563:Stable 27513:Landau 27487:Gumbel 27441:Cauchy 27369:Pareto 27181:Erlang 27162:Scaled 27117:Benini 26956:Panjer 26734: 26655: 26647: 26637: 26572: 26531: 26478: 26407: 26384: 26345: 26306: 26296: 26247: 26187: 26162: 26064: 26028: 25981: 25924: 25914: 25875: 25865: 25824: 25814: 25806: 25767: 25757: 25749: 25695: 25687: 25650:429469 25648: 25613: 25605: 25564: 25556: 25508: 25498: 25413: 25403: 25395: 25353: 25307: 25280: 25237: 25227: 25219: 24995: 24987: 24897: 24759: 24649: 24614: 24525: 24515: 24476:1 July 24457: 24425: 24415: 24384: 24374: 24311: 24301: 24224: 24194: 24184: 23970: 23960: 23934: 23823: 23790: 23780: 23741:(NIST) 23666: 22263:Fading 22001:where 21921:income 21775:PacBio 20232:) as: 19824:has a 18129:) as: 17696:has a 15252:where 14927:Suzuki 12412: 12119:] and 11900: 11780: 11635: 9999:median 8818:where 8395: 8304:where 7985:median 6801:moment 6793:, the 6007:where 4860:where 4684: 4675: 4666: 4650: 4630: 4624: 4616:where 4499: 4478: 4470: 4439: 4416: 4411: 4405: 4394: 4388: 4366: 4360: 4354: 4335: 4317: 4281: 4263: 4174: 4156: 4101: 4077: 4049: 4008: 3996: 3968: 3890: 3884: 3872: 3859: 3839: 3833: 3813: 3807: 3715: 3702: 3662: 3656: 3636: 3628: 3609: 3589: 3569: 3563: 3535: 3522: 3510:median 3494: 3481: 3461: 3435: 3415: 3389: 3364: 3338: 3320: 3284: 3264: 3254: 3249: 3211: 3174: 3151: 3133: 3063:where 3048: 3035: 3015: 3002: 2979: 2961: 2941: 2916: 2896: 2871: 2847: 2828: 2808: 2787: 2760: 2754: 2734:) and 2556: 2550: 2485:Gibrat 2444:= exp( 2401:whose 2260: 2232: 2184: 2179: 2171: 2145: 2099: 2068: 2060: 2048: 2019: 2014: 2011: 1993: 1968: 1944: 1934: 1929: 1917: 1888: 1883: 1880: 1862: 1854: 1849: 1808: 1780: 1766: 1753: 1727: 1714: 1697: 1665: 1623: 1604: 1588: 1579: 1552: 1511: 1503: 1475: 1469: 1464: 1453: 1432: 1404: 1366: 1342: 1325: 1301: 1284: 1278: 1212: 1179: 1171: 1143: 1119: 1102: 1094: 1063: 1055: 1027: 1019: 1000: 986: 958: 952: 946: 934: 923:Median 906: 898: 872: 858: 766: 690: 604: 591: 573: 540: 534: 528: 467: 427: 421: 416: 399: 390: 362: 356: 341: 329: 296: 284: 259: 253: 235: 223: 195: 187: 167: 153: 126: 114: 60: 54: 27760:Ewens 27586:Voigt 27558:Slash 27339:Lomax 27334:Log-t 27239:Gamma 27186:Hyper 27176:Davis 27171:Dagum 27027:Bates 27017:ARGUS 26901:Borel 26589:arXiv 26529:S2CID 26503:arXiv 26343:JSTOR 26276:arXiv 26245:S2CID 26227:arXiv 26160:S2CID 26115:arXiv 26048:(PDF) 25562:S2CID 25305:JSTOR 25278:S2CID 25191:arXiv 24985:JSTOR 24916:arXiv 24877:arXiv 24815:(PDF) 24222:JSTOR 24164:arXiv 24099:(PDF) 24055:(PDF) 23903:(PDF) 23896:(PDF) 23876:(PDF) 23869:(PDF) 23842:(PDF) 23809:(PDF) 23675:(PDF) 23664:S2CID 23646:arXiv 23632:(PDF) 22272:Notes 21735:chess 21611:2.718 18527:with 17051:with 16926:with 14862:with 14619:then 12962:is a 12769:Other 12000:and 11861:then 11739:then 11591:then 7958:E = e 5768:with 4930:is a 2568:be a 2419:= ln( 2397:of a 310:scale 28009:and 27967:Kent 27394:Rice 27309:Lévy 27137:Burr 27067:PERT 27032:Beta 26981:Zeta 26873:Zipf 26790:list 26736:5735 26732:SSRN 26635:ISBN 26570:ISSN 26476:ISSN 26405:ISBN 26382:PMID 26304:PMID 26185:ISBN 26062:ISBN 26026:ISSN 25979:ISSN 25922:PMID 25873:PMID 25822:PMID 25804:ISSN 25765:PMID 25747:ISSN 25693:PMID 25685:ISSN 25646:PMID 25611:PMID 25603:ISSN 25554:PMID 25506:OCLC 25496:ISBN 25477:2018 25411:PMID 25393:ISSN 25351:ISSN 25235:PMID 25217:ISSN 25046:html 24993:PMID 24895:ISBN 24787:2009 24757:ISSN 24745:1977 24647:PMID 24612:PMID 24523:PMID 24478:2017 24455:ISBN 24449:help 24413:ISBN 24382:PMID 24309:PMID 24192:PMID 23968:OCLC 23958:ISBN 23932:ISBN 23778:ISBN 22404:and 22204:The 21832:and 21773:The 19792:and 18955:and 18720:and 18642:and 18085:for 17136:for 16785:for 16160:and 16140:for 16065:and 15601:and 15554:and 15530:and 15123:and 13242:and 13152:Let 12612:and 12295:are 11935:for 11671:> 11660:for 11452:and 11190:and 9700:mean 8936:< 8850:and 8702:> 8618:> 8600:> 8524:> 8169:are 8057:Mode 8001:mode 7999:The 7989:mode 7987:and 7981:mean 7960:for 7593:and 6799:-th 6688:GVar 6428:GVar 6293:The 6218:> 6144:> 5543:The 4863:erfc 4796:erfc 4526:The 3825:and 3799:Let 3427:and 3276:and 3074:< 2732:mean 2730:(or 2706:and 2623:> 2592:and 2542:Let 2487:and 2385:(or 2381:, a 975:Mode 847:Mean 290:> 27845:LKJ 27142:Chi 26707:doi 26684:doi 26653:Zbl 26560:doi 26521:doi 26468:doi 26456:238 26374:doi 26335:doi 26294:PMC 26286:doi 26237:doi 26152:doi 26125:doi 26018:doi 25969:hdl 25961:doi 25912:PMC 25904:doi 25863:PMC 25853:doi 25812:PMC 25796:doi 25792:103 25755:PMC 25739:doi 25677:doi 25638:doi 25593:doi 25546:doi 25401:PMC 25385:doi 25341:doi 25270:doi 25258:155 25225:PMC 25209:doi 25160:doi 25123:doi 25043:pdf 25020:doi 24977:doi 24945:doi 24887:doi 24854:doi 24827:doi 24791:doi 24749:doi 24722:doi 24718:116 24695:doi 24639:doi 24604:doi 24600:108 24575:doi 24513:PMC 24505:doi 24423:PMC 24405:doi 24372:PMC 24364:doi 24299:PMC 24289:doi 24253:doi 24182:PMC 24174:doi 24111:doi 24075:hdl 24067:doi 24033:doi 23998:doi 23922:doi 23831:doi 23817:150 23656:doi 23642:299 22233:in 22186:In 22179:In 22152:In 22112:In 22009:erf 21968:erf 21915:In 21863:In 21791:max 21032:MLE 19840:). 19834:CLT 17706:CLT 17060:sem 17020:sem 16928:n-1 16805:is 16116:Var 15934:by 14970:CDF 14818:If 14784:Var 14760:Var 14568:If 14360:cor 14237:cor 13034:If 12974:exp 12910:If 12781:). 12635:var 11323:LN2 11285:LN7 11242:exp 11211:exp 11152:exp 11087:LN7 11063:LN2 10853:exp 10601:exp 10505:, σ 10427:exp 10255:exp 10067:exp 9923:exp 9772:exp 9638:or 8354:Med 7965:≥ 1 7890:Var 7730:Var 7274:Var 7056:Var 6555:). 6484:GCV 6367:GSD 6359:is 6241:sin 6202:sin 6135:sin 6109:0.5 6063:a. 5817:exp 5141:Var 5130:is 5030:is 4955:exp 4878:If 4731:erf 4696:). 4530:is 4422:exp 2920:log 2875:log 2744:not 2523:ln( 2471:or 2434:of 2405:is 2377:In 2194:erf 2154:erf 1996:Var 1979:log 1865:Var 1817:log 1528:MGF 1436:log 1369:exp 1328:exp 1287:exp 1222:exp 1182:exp 1105:exp 1066:exp 989:exp 937:exp 861:exp 790:exp 728:erf 693:exp 560:erf 517:CDF 430:exp 379:PDF 275:), 28149:: 26730:. 26726:. 26703:18 26701:. 26680:51 26678:. 26674:. 26651:, 26645:MR 26643:, 26568:. 26556:27 26554:. 26550:. 26527:. 26519:. 26511:. 26499:85 26497:. 26474:. 26466:. 26454:. 26450:. 26427:. 26380:. 26370:15 26368:. 26364:. 26341:. 26331:94 26329:. 26325:. 26302:. 26292:. 26284:. 26270:. 26266:. 26243:. 26235:. 26221:. 26158:. 26148:81 26146:. 26123:. 26060:. 26024:. 26016:. 26004:. 26000:. 25977:. 25967:. 25959:. 25949:20 25947:. 25943:. 25920:. 25910:. 25900:33 25898:. 25894:. 25871:. 25861:. 25847:. 25843:. 25820:. 25810:. 25802:. 25790:. 25786:. 25763:. 25753:. 25745:. 25733:. 25729:. 25691:. 25683:. 25671:. 25667:. 25644:. 25634:32 25632:. 25609:. 25601:. 25589:29 25587:. 25583:. 25560:. 25552:. 25542:25 25540:. 25504:. 25468:. 25423:^ 25409:. 25399:. 25391:. 25381:15 25379:. 25375:. 25363:^ 25349:. 25337:51 25335:. 25331:. 25317:^ 25301:32 25299:. 25276:. 25268:. 25256:. 25233:. 25223:. 25215:. 25207:. 25199:. 25187:56 25185:. 25181:. 25158:. 25148:18 25146:. 25121:. 25111:19 25109:. 25105:. 25093:^ 25032:^ 25014:. 24991:. 24983:. 24973:36 24971:. 24939:. 24893:. 24885:. 24850:46 24848:. 24823:78 24821:. 24817:. 24785:. 24781:. 24769:^ 24755:. 24743:. 24716:. 24691:81 24689:. 24645:. 24635:65 24633:. 24610:. 24598:. 24571:51 24569:. 24565:. 24535:^ 24521:. 24511:. 24501:32 24499:. 24495:. 24441:: 24439:}} 24435:{{ 24421:. 24411:. 24380:. 24370:. 24360:73 24358:. 24354:. 24336:, 24307:. 24297:. 24283:. 24279:. 24267:^ 24249:51 24247:. 24243:. 24218:35 24216:. 24204:^ 24190:. 24180:. 24172:. 24160:21 24158:. 24154:. 24140:^ 24107:32 24105:. 24101:. 24073:. 24063:75 24061:. 24057:. 24031:. 24021:66 24019:. 23994:18 23992:. 23980:^ 23966:. 23930:, 23829:. 23815:. 23811:. 23788:MR 23786:, 23776:, 23754:^ 23733:. 23722:^ 23706:. 23690:^ 23670:. 23662:. 23654:. 23640:. 23634:. 21927:). 21794:, 17920:. 17712:: 17257:ln 17199:ln 16290:ln 16194:ln 16180:: 15974:. 15804:ln 15737:ln 15701:, 15673:ln 15648:ln 15629:ln 15493:ln 15468:ln 15449:ln 15405:ln 15221:ln 14961:). 14752:, 14690:. 14502:ln 13909:ln 13772:ln 13577:ln 13391:ln 13093:ln 12765:. 12758:. 12723:ln 12643:ln 12578:ln 11949:0. 11674:0. 11623:ln 11528:. 11466:ln 11367:ln 11274:. 10971:ln 10889:ln 10877:ln 10791:ln 10707:,σ 10654:ln 10619:ln 10566:ln 10454:ln 10307:ln 10273:ln 10209:ln 10085:ln 9943:ln 9792:ln 9550:ln 9499:ln 9436:ln 9372:ln 9231:ln 9188:ln 9054:ln 8998:ln 8858:. 8787:ln 8131:ln 8014:ln 7983:, 7870:ln 7801:ln 7693:ln 7624:ln 7523:CV 7467:SD 7431:CV 7250:SD 6779:. 6668:GM 6539:CV 6476:, 6417:. 6305:GM 6285:) 6273:) 6176:ln 6121:. 6051:. 5418:ln 5300:. 4870:. 4813:ln 4744:ln 4576:ln 4445:ln 4320:ln 4266:ln 4210:ln 4159:ln 4092:ln 4080:ln 3740:ln 3300:ln 3183:ln 3095:. 2742:, 2491:. 2483:, 2460:, 2438:, 2389:) 1540:CF 634:ln 576:ln 452:ln 27857:t 27818:t 27686:q 27678:q 27670:q 27662:κ 27653:κ 27644:κ 27635:κ 27626:κ 27570:t 27537:t 27506:U 27504:S 27466:q 27453:z 27285:T 27216:F 26792:) 26788:( 26778:e 26771:t 26764:v 26738:. 26728:7 26713:. 26709:: 26692:. 26686:: 26597:. 26591:: 26576:. 26562:: 26535:. 26523:: 26515:: 26505:: 26482:. 26470:: 26462:: 26413:. 26388:. 26376:: 26337:: 26310:. 26288:: 26278:: 26272:8 26251:. 26239:: 26229:: 26223:8 26193:. 26166:. 26154:: 26131:. 26127:: 26117:: 26070:. 26032:. 26020:: 26012:: 26006:5 25985:. 25971:: 25963:: 25955:: 25928:. 25906:: 25879:. 25855:: 25849:6 25828:. 25798:: 25771:. 25741:: 25735:4 25699:. 25679:: 25673:7 25652:. 25640:: 25617:. 25595:: 25568:. 25548:: 25512:. 25479:. 25437:. 25417:. 25387:: 25357:. 25343:: 25311:. 25284:. 25272:: 25264:: 25241:. 25211:: 25203:: 25193:: 25166:. 25162:: 25154:: 25131:. 25125:: 25117:: 25026:. 25022:: 25016:7 24999:. 24979:: 24951:. 24947:: 24941:6 24924:. 24918:: 24903:. 24889:: 24879:: 24860:. 24856:: 24833:. 24829:: 24799:. 24793:: 24763:. 24751:: 24728:. 24724:: 24701:. 24697:: 24653:. 24641:: 24618:. 24606:: 24583:. 24577:: 24529:. 24507:: 24480:. 24451:) 24431:; 24407:: 24388:. 24366:: 24315:. 24291:: 24285:7 24261:. 24255:: 24228:. 24198:. 24176:: 24166:: 24119:. 24113:: 24083:. 24077:: 24069:: 24039:. 24035:: 24027:: 24004:. 24000:: 23974:. 23924:: 23851:. 23833:: 23748:. 23716:. 23658:: 23648:: 23590:) 23587:1 23579:2 23575:n 23571:( 23568:2 23562:4 23557:2 23553:S 23547:+ 23541:) 23538:1 23530:1 23526:n 23522:( 23519:2 23513:4 23508:1 23504:S 23498:+ 23491:2 23487:n 23481:2 23476:2 23472:S 23466:+ 23459:1 23455:n 23449:2 23444:1 23440:S 23433:2 23427:] 23422:) 23417:2 23412:2 23408:S 23399:2 23394:1 23390:S 23386:( 23381:2 23378:1 23373:+ 23370:) 23365:2 23343:1 23326:( 23322:e 23318:[ 23314:= 23303:) 23296:) 23293:1 23285:2 23281:n 23277:( 23274:2 23268:4 23263:2 23253:+ 23247:) 23244:1 23236:1 23232:n 23228:( 23225:2 23219:4 23214:1 23204:+ 23197:2 23193:n 23187:2 23182:2 23172:+ 23165:1 23161:n 23155:2 23150:1 23139:( 23133:2 23127:] 23121:) 23116:2 23112:X 23108:( 23099:E 23091:) 23086:1 23082:X 23078:( 23069:E 23060:[ 23056:= 23046:] 23040:) 23035:2 23031:X 23027:( 23024:E 23019:) 23014:1 23010:X 23006:( 23003:E 22997:[ 22962:1 22958:p 22954:= 22949:) 22944:1 22940:p 22936:( 22933:g 22930:o 22927:l 22923:e 22919:= 22916:] 22911:1 22901:p 22894:[ 22891:E 22869:) 22866:n 22858:1 22854:p 22850:( 22846:/ 22842:) 22839:1 22836:p 22830:1 22827:( 22821:2 22817:/ 22813:1 22810:+ 22807:) 22802:1 22798:p 22794:( 22791:g 22788:o 22785:l 22781:e 22777:= 22774:] 22769:1 22759:p 22752:[ 22749:E 22727:1 22717:p 22691:) 22688:) 22685:n 22677:1 22667:p 22660:( 22656:/ 22652:) 22647:1 22637:p 22627:1 22624:( 22618:) 22613:1 22603:p 22596:( 22593:g 22590:o 22587:l 22583:e 22579:: 22576:) 22571:1 22567:p 22563:( 22560:I 22557:C 22537:) 22534:) 22531:n 22523:1 22519:p 22515:( 22511:/ 22507:) 22504:1 22501:p 22495:1 22492:( 22489:, 22486:) 22481:1 22477:p 22473:( 22470:g 22467:o 22464:l 22461:( 22458:N 22443:) 22438:1 22428:p 22421:( 22418:g 22415:o 22412:l 22392:) 22389:n 22385:/ 22381:) 22378:1 22375:p 22369:1 22366:( 22361:1 22357:p 22353:, 22348:1 22344:p 22340:( 22337:N 22320:1 22310:p 22216:. 22201:. 22149:. 22097:) 22094:x 22091:( 22068:1 22061:) 22055:2 22046:( 22039:2 22036:= 22033:G 21988:) 21983:2 21975:( 21965:= 21962:G 21903:. 21886:. 21836:. 21816:. 21802:. 21789:C 21764:. 21675:6 21668:/ 21663:1 21660:= 21608:= 21605:e 21579:6 21575:1 21570:= 21534:σ 21514:) 21507:) 21504:1 21496:2 21492:n 21488:( 21485:2 21479:4 21474:2 21464:+ 21458:) 21455:1 21447:1 21443:n 21439:( 21436:2 21430:4 21425:1 21415:+ 21408:2 21404:n 21398:2 21393:2 21383:+ 21376:1 21372:n 21366:2 21361:1 21350:( 21344:2 21341:1 21336:+ 21333:) 21328:2 21323:2 21310:2 21305:1 21297:( 21292:2 21289:1 21284:+ 21281:) 21276:2 21263:1 21255:( 21251:e 21247:= 21243:] 21238:) 21233:2 21228:2 21224:S 21215:2 21210:1 21206:S 21202:( 21197:2 21194:1 21189:+ 21186:) 21181:2 21159:1 21142:( 21138:e 21134:[ 21130:E 21127:= 21123:] 21117:) 21112:2 21108:X 21104:( 21095:E 21087:) 21082:1 21078:X 21074:( 21065:E 21056:[ 21052:E 21008:) 20999:) 20996:1 20988:2 20984:n 20980:( 20977:2 20971:4 20966:2 20962:S 20956:+ 20950:) 20947:1 20939:1 20935:n 20931:( 20928:2 20922:4 20917:1 20913:S 20907:+ 20900:2 20896:n 20890:2 20885:2 20881:S 20875:+ 20868:1 20864:n 20858:2 20853:1 20849:S 20837:2 20826:1 20822:z 20815:) 20810:2 20805:2 20801:S 20795:2 20792:1 20782:2 20777:1 20773:S 20767:2 20764:1 20759:+ 20754:2 20732:1 20715:( 20711:( 20706:e 20702:: 20698:) 20688:2 20683:2 20678:2 20668:+ 20663:2 20654:e 20646:2 20641:2 20636:1 20626:+ 20621:1 20612:e 20606:= 20600:) 20595:2 20591:X 20587:( 20584:E 20579:) 20574:1 20570:X 20566:( 20563:E 20556:( 20552:I 20549:C 20524:) 20521:1 20513:2 20509:n 20505:( 20502:2 20496:4 20491:2 20487:S 20481:+ 20475:) 20472:1 20464:1 20460:n 20456:( 20453:2 20447:4 20442:1 20438:S 20432:+ 20425:2 20421:n 20415:2 20410:2 20406:S 20400:+ 20393:1 20389:n 20383:2 20378:1 20374:S 20362:2 20351:1 20347:z 20340:) 20335:2 20330:2 20326:S 20320:2 20317:1 20307:2 20302:1 20298:S 20292:2 20289:1 20284:+ 20279:2 20257:1 20240:( 20209:) 20202:) 20199:1 20191:2 20187:n 20183:( 20180:2 20174:4 20169:2 20159:+ 20153:) 20150:1 20142:1 20138:n 20134:( 20131:2 20125:4 20120:1 20110:+ 20103:2 20099:n 20093:2 20088:2 20078:+ 20071:1 20067:n 20061:2 20056:1 20046:, 20043:) 20038:2 20033:2 20020:2 20015:1 20007:( 20002:2 19999:1 19994:+ 19991:) 19986:2 19973:1 19965:( 19961:( 19957:N 19951:) 19946:2 19941:2 19937:S 19931:2 19928:1 19918:2 19913:1 19909:S 19903:2 19900:1 19895:+ 19890:2 19868:1 19851:( 19810:2 19805:2 19801:S 19778:2 19773:1 19769:S 19746:2 19724:1 19680:) 19671:) 19668:1 19660:2 19656:n 19652:( 19649:2 19643:4 19638:2 19634:S 19628:+ 19622:) 19619:1 19611:1 19607:n 19603:( 19600:2 19594:4 19589:1 19585:S 19579:+ 19572:2 19568:n 19562:2 19557:2 19553:S 19547:+ 19540:1 19536:n 19530:2 19525:1 19521:S 19509:2 19498:1 19494:z 19487:) 19482:2 19477:2 19473:S 19467:2 19464:1 19454:2 19449:1 19445:S 19439:2 19436:1 19431:+ 19426:2 19404:1 19387:( 19383:( 19378:e 19374:: 19370:) 19360:2 19355:2 19350:2 19340:+ 19335:2 19326:e 19318:2 19313:2 19308:1 19298:+ 19293:1 19284:e 19278:= 19272:) 19267:2 19263:X 19259:( 19256:E 19251:) 19246:1 19242:X 19238:( 19235:E 19228:( 19224:I 19221:C 19195:) 19190:2 19185:2 19172:2 19167:1 19159:( 19154:2 19151:1 19146:+ 19143:) 19138:2 19125:1 19117:( 19113:e 19109:= 19100:2 19095:2 19090:2 19080:+ 19075:2 19066:e 19058:2 19053:2 19048:1 19038:+ 19033:1 19024:e 19018:= 19012:) 19007:2 19003:X 18999:( 18996:E 18991:) 18986:1 18982:X 18978:( 18975:E 18934:) 18926:n 18921:2 18916:2 18912:S 18906:+ 18901:n 18896:2 18891:1 18887:S 18875:2 18864:1 18860:z 18851:2 18829:1 18811:( 18806:e 18802:: 18799:) 18792:2 18779:1 18770:e 18766:( 18763:I 18760:C 18733:2 18729:n 18706:1 18702:n 18681:) 18676:2 18671:2 18663:, 18658:2 18650:( 18630:) 18625:2 18620:1 18612:, 18607:1 18599:( 18563:2 18552:1 18549:, 18546:1 18540:n 18536:t 18511:2 18500:1 18496:z 18461:) 18452:) 18449:1 18443:n 18440:( 18437:2 18431:4 18427:S 18421:+ 18416:n 18411:2 18407:S 18395:2 18384:1 18380:z 18371:2 18366:2 18362:S 18356:+ 18340:( 18335:e 18331:: 18327:) 18319:2 18314:2 18304:+ 18297:e 18293:= 18290:) 18287:X 18284:( 18281:E 18277:( 18273:I 18270:C 18245:) 18242:1 18236:n 18233:( 18230:2 18224:4 18220:S 18214:+ 18209:n 18204:2 18200:S 18188:2 18177:1 18173:z 18164:2 18159:2 18155:S 18149:+ 18111:2 18106:2 18096:+ 18068:) 18061:) 18058:1 18052:n 18049:( 18046:2 18040:4 18030:+ 18025:n 18020:2 18010:, 18005:2 18000:2 17990:+ 17983:( 17979:N 17962:2 17957:2 17953:S 17947:+ 17907:) 17900:) 17897:1 17891:n 17888:( 17885:2 17879:4 17869:, 17864:2 17859:2 17848:( 17844:N 17827:2 17822:2 17818:S 17794:) 17787:1 17781:n 17774:4 17766:2 17760:, 17755:2 17746:( 17742:N 17725:2 17721:S 17682:2 17678:S 17655:) 17649:n 17644:2 17634:, 17627:( 17623:N 17555:2 17550:2 17540:+ 17533:e 17529:= 17526:) 17523:X 17520:( 17517:E 17490:) 17481:) 17478:1 17472:n 17469:( 17466:2 17460:4 17456:S 17450:+ 17445:n 17440:2 17436:S 17424:2 17413:1 17409:z 17400:2 17395:2 17391:S 17385:+ 17369:( 17364:e 17360:: 17357:) 17354:) 17351:X 17348:( 17345:E 17342:( 17339:I 17336:C 17307:1 17301:n 17294:2 17289:) 17268:i 17264:x 17253:( 17246:i 17235:= 17230:2 17226:S 17221:, 17216:n 17210:i 17206:x 17194:i 17183:= 17109:n 17103:/ 17099:1 17095:) 17072:( 17069:= 17039:] 17034:q 17030:) 17016:( 17012:/ 16978:[ 16952:e 16948:= 16920:q 16904:n 16898:/ 16882:= 16878:e 16875:s 16854:] 16844:e 16841:s 16831:q 16813:[ 16776:e 16758:] 16753:2 16749:) 16735:( 16731:/ 16706:[ 16703:= 16700:] 16695:2 16691:) 16677:( 16661:, 16656:2 16652:) 16638:( 16634:/ 16620:[ 16600:] 16586:/ 16561:[ 16558:= 16555:] 16529:, 16515:/ 16501:[ 16481:] 16475:2 16472:+ 16466:, 16460:2 16451:[ 16431:] 16425:+ 16419:, 16407:[ 16357:. 16353:) 16347:2 16337:x 16329:/ 16322:2 16304:+ 16301:1 16297:( 16287:= 16282:2 16273:, 16269:) 16261:2 16251:x 16243:/ 16237:2 16220:+ 16217:1 16208:x 16201:( 16191:= 16128:] 16125:X 16122:[ 16096:] 16093:X 16090:[ 16084:E 16070:s 16047:x 16022:n 16018:x 16014:, 16008:, 16003:2 15999:x 15995:, 15990:1 15986:x 15960:2 15936:n 15932:n 15868:n 15852:. 15847:n 15841:2 15836:) 15815:i 15811:x 15800:( 15793:i 15782:= 15777:2 15759:, 15754:n 15748:i 15744:x 15732:i 15721:= 15689:) 15684:n 15680:x 15670:, 15664:, 15659:2 15655:x 15645:, 15640:1 15636:x 15567:N 15532:σ 15528:μ 15512:. 15509:) 15504:n 15500:x 15490:, 15484:, 15479:2 15475:x 15465:, 15460:1 15456:x 15440:, 15434:( 15429:N 15421:+ 15416:i 15412:x 15400:i 15389:= 15386:) 15381:n 15377:x 15373:, 15367:, 15362:2 15358:x 15354:, 15349:1 15345:x 15335:, 15329:( 15306:) 15301:2 15293:, 15287:( 15282:N 15240:, 15237:) 15232:i 15228:x 15218:( 15210:, 15195:i 15191:x 15187:1 15180:n 15175:1 15172:= 15169:i 15161:= 15158:) 15152:, 15146:( 15143:L 15125:σ 15121:μ 15101:. 15085:. 15080:1 15072:] 15068:1 15065:+ 15060:) 15055:3 15047:( 15043:/ 15034:) 15029:x 15020:e 15014:( 15008:[ 15003:= 15000:) 14994:, 14988:; 14985:x 14982:( 14979:F 14957:( 14945:) 14939:, 14933:( 14921:X 14901:) 14896:2 14888:, 14882:( 14870:Y 14850:) 14847:Y 14844:( 14832:Y 14826:X 14815:. 14808:. 14796:] 14793:X 14790:[ 14781:= 14778:] 14775:c 14772:+ 14769:X 14766:[ 14740:c 14737:+ 14734:] 14731:X 14728:[ 14722:E 14719:= 14716:] 14713:c 14710:+ 14707:X 14704:[ 14698:E 14678:) 14672:+ 14669:, 14666:c 14663:( 14657:x 14633:c 14630:+ 14627:X 14607:) 14602:2 14594:, 14588:( 14576:X 14549:2 14545:/ 14539:2 14534:Z 14522:) 14517:+ 14513:S 14509:( 14499:= 14490:Z 14476:2 14472:/ 14466:2 14461:j 14453:+ 14448:j 14439:e 14433:2 14429:/ 14423:2 14418:i 14410:+ 14405:i 14396:e 14390:j 14380:i 14367:j 14364:i 14354:j 14351:, 14348:i 14337:2 14332:+ 14328:S 14323:/ 14319:1 14316:= 14313:] 14308:j 14304:X 14300:[ 14294:E 14291:] 14286:i 14282:X 14278:[ 14272:E 14267:j 14257:i 14244:j 14241:i 14231:j 14228:, 14225:i 14214:2 14209:+ 14205:S 14200:/ 14196:1 14193:= 14184:2 14179:Z 14165:2 14161:/ 14155:2 14150:i 14142:+ 14137:i 14128:e 14122:i 14114:= 14111:] 14106:i 14102:X 14098:[ 14092:E 14087:i 14079:= 14075:] 14069:i 14065:X 14059:i 14050:[ 14043:E 14040:= 14031:+ 14027:S 13988:. 13983:2 13978:2 13973:Z 13958:2 13953:2 13943:+ 13939:] 13931:j 13922:e 13914:[ 13906:= 13897:Z 13885:, 13881:] 13877:1 13874:+ 13866:2 13862:) 13854:j 13845:e 13838:( 13829:j 13821:2 13817:e 13807:) 13804:1 13794:2 13785:e 13781:( 13777:[ 13769:= 13760:2 13755:Z 13723:= 13718:j 13691:j 13687:X 13662:. 13657:2 13652:2 13647:Z 13633:] 13627:2 13623:/ 13617:2 13612:j 13604:+ 13599:j 13590:e 13582:[ 13574:= 13565:Z 13553:, 13549:] 13545:1 13542:+ 13534:2 13530:) 13524:2 13520:/ 13514:2 13509:j 13501:+ 13496:j 13487:e 13480:( 13475:) 13472:1 13462:2 13457:j 13448:e 13444:( 13437:2 13432:j 13424:+ 13419:j 13411:2 13407:e 13396:[ 13388:= 13379:2 13374:Z 13344:Z 13324:Y 13302:j 13298:X 13292:n 13287:1 13284:= 13281:j 13273:= 13270:Y 13210:) 13205:2 13200:j 13192:, 13187:j 13179:( 13165:j 13161:X 13137:) 13132:2 13124:, 13118:( 13113:N 13105:) 13102:X 13099:( 13073:) 13068:2 13060:, 13054:( 13042:X 13020:. 13017:) 13012:2 13004:, 12998:( 12986:) 12983:X 12980:( 12950:) 12945:2 12937:, 12931:( 12926:N 12918:X 12878:. 12873:A 12868:2 12864:G 12858:= 12855:H 12832:A 12812:G 12792:H 12742:) 12737:i 12733:X 12729:( 12698:2 12673:n 12669:/ 12665:] 12662:) 12657:i 12653:X 12649:( 12640:[ 12630:= 12625:2 12600:] 12597:) 12592:i 12588:X 12584:( 12575:[ 12572:E 12569:= 12540:n 12518:i 12514:X 12493:n 12458:. 12453:) 12446:2 12441:j 12431:n 12426:1 12423:= 12420:j 12409:, 12404:j 12394:n 12389:1 12386:= 12383:j 12372:( 12356:j 12352:X 12346:n 12341:1 12338:= 12335:j 12326:= 12323:Y 12303:n 12283:) 12278:2 12273:j 12265:, 12260:j 12252:( 12238:j 12234:X 12210:n 12188:2 12183:2 12175:+ 12170:2 12165:1 12157:= 12152:2 12105:2 12092:1 12084:= 12073:[ 12059:2 12051:+ 12046:1 12038:= 12013:2 12009:X 11986:1 11982:X 11943:a 11923:) 11918:2 11908:2 11904:a 11897:, 11891:a 11888:( 11874:a 11870:X 11849:) 11844:2 11836:, 11830:( 11818:X 11796:. 11793:) 11788:2 11777:, 11768:( 11753:X 11750:1 11727:) 11722:2 11714:, 11708:( 11696:X 11668:a 11648:) 11643:2 11632:, 11629:a 11620:+ 11614:( 11602:X 11599:a 11579:) 11574:2 11566:, 11560:( 11548:X 11516:) 11511:2 11506:N 11497:/ 11491:2 11486:N 11478:+ 11475:1 11472:( 11463:= 11460:v 11439:) 11431:2 11426:N 11417:/ 11411:2 11406:N 11398:+ 11395:1 11389:/ 11383:N 11374:( 11364:= 11341:) 11338:v 11335:, 11329:( 11317:) 11312:N 11304:, 11299:N 11291:( 11260:1 11254:) 11251:v 11248:( 11237:) 11234:2 11230:/ 11226:v 11223:+ 11217:( 11208:= 11203:N 11178:) 11175:2 11171:/ 11167:v 11164:+ 11158:( 11149:= 11144:N 11119:) 11114:N 11106:, 11101:N 11093:( 11081:) 11078:v 11075:, 11069:( 11028:) 11021:) 11016:2 11011:N 11002:/ 10996:2 10991:N 10983:+ 10980:1 10977:( 10968:2 10961:2 10955:] 10944:2 10939:N 10930:/ 10924:2 10919:N 10911:+ 10908:1 10902:N 10883:x 10872:[ 10860:( 10844:) 10838:2 10833:N 10824:/ 10818:2 10813:N 10805:+ 10802:1 10798:( 10785:2 10780:x 10776:1 10771:= 10768:) 10762:N 10753:, 10747:N 10738:; 10735:x 10732:( 10729:P 10717:N 10713:N 10709:N 10705:N 10686:] 10679:) 10674:g 10666:( 10658:2 10650:2 10645:) 10642:m 10638:/ 10634:x 10631:( 10623:2 10608:[ 10590:2 10585:) 10580:g 10572:( 10563:x 10559:1 10554:= 10551:) 10545:g 10536:, 10532:m 10528:; 10525:x 10522:( 10519:P 10507:g 10499:g 10480:] 10474:2 10470:) 10460:x 10451:( 10446:2 10434:[ 10422:x 10419:1 10407:2 10397:= 10394:) 10386:, 10378:; 10375:x 10372:( 10369:P 10342:] 10335:) 10332:1 10329:+ 10324:2 10320:v 10316:c 10313:( 10304:2 10299:) 10296:m 10292:/ 10288:x 10285:( 10277:2 10262:[ 10244:2 10237:) 10234:1 10231:+ 10226:2 10222:v 10218:c 10215:( 10204:x 10200:1 10195:= 10192:) 10188:v 10185:c 10181:, 10177:m 10173:; 10170:x 10167:( 10164:P 10133:] 10124:2 10116:2 10111:) 10108:m 10104:/ 10100:x 10097:( 10089:2 10074:[ 10056:2 10048:x 10044:1 10039:= 10036:) 10028:, 10024:m 10020:; 10017:x 10014:( 10011:P 9980:] 9973:v 9970:2 9963:2 9959:) 9949:x 9940:( 9930:[ 9912:2 9905:v 9900:x 9896:1 9891:= 9888:) 9884:v 9880:, 9872:; 9869:x 9866:( 9863:P 9836:] 9827:2 9819:2 9812:2 9808:) 9798:x 9789:( 9779:[ 9761:2 9753:x 9749:1 9744:= 9741:) 9733:, 9725:; 9722:x 9719:( 9716:P 9656:, 9623:, 9584:] 9569:) 9564:1 9560:k 9556:( 9544:[ 9533:] 9518:) 9513:2 9509:k 9505:( 9493:[ 9483:] 9472:2 9455:) 9450:1 9446:k 9442:( 9430:[ 9419:] 9408:2 9391:) 9386:2 9382:k 9378:( 9366:[ 9349:2 9344:2 9334:+ 9327:e 9323:= 9316:] 9313:] 9308:2 9304:k 9300:, 9295:1 9291:k 9287:[ 9281:X 9275:X 9272:[ 9269:E 9258:] 9243:) 9240:k 9237:( 9225:[ 9215:1 9209:] 9200:) 9197:k 9194:( 9180:2 9172:+ 9163:[ 9146:2 9141:2 9131:+ 9124:e 9120:= 9113:] 9110:k 9104:X 9098:X 9095:[ 9092:E 9081:] 9066:) 9063:k 9060:( 9048:[ 9038:] 9027:2 9010:) 9007:k 9004:( 8992:[ 8975:2 8970:2 8960:+ 8953:e 8949:= 8942:] 8939:k 8933:X 8927:X 8924:[ 8921:E 8894:k 8874:X 8802:) 8793:k 8779:2 8771:+ 8762:( 8749:2 8738:2 8735:1 8729:+ 8722:e 8718:= 8715:x 8712:d 8708:) 8705:k 8699:X 8693:x 8690:( 8685:X 8681:f 8677:x 8667:k 8659:= 8656:) 8653:k 8650:( 8647:g 8624:) 8621:k 8615:X 8612:( 8609:P 8606:] 8603:k 8597:X 8591:X 8588:[ 8582:E 8579:= 8576:) 8573:k 8570:( 8567:g 8540:. 8537:x 8534:d 8530:) 8527:k 8521:X 8515:x 8512:( 8507:X 8503:f 8499:x 8489:k 8481:= 8478:) 8475:k 8472:( 8469:g 8446:k 8426:X 8398:. 8382:= 8373:e 8369:= 8366:] 8363:X 8360:[ 8328:) 8322:( 8313:q 8289:, 8284:) 8278:( 8269:q 8264:) 8250:( 8237:= 8232:) 8226:( 8217:q 8210:+ 8203:e 8199:= 8196:) 8190:( 8185:X 8181:q 8157:X 8137:X 8128:= 8125:Y 8098:. 8091:2 8076:e 8072:= 8069:] 8066:X 8063:[ 8034:0 8031:= 8024:) 8020:f 8011:( 7995:. 7963:n 7937:. 7933:) 7924:2 7920:] 7916:X 7913:[ 7907:E 7902:] 7899:X 7896:[ 7884:+ 7881:1 7877:( 7867:= 7863:) 7855:2 7851:] 7847:X 7844:[ 7838:E 7833:] 7828:2 7824:X 7820:[ 7814:E 7808:( 7798:= 7789:2 7777:, 7773:) 7765:2 7761:] 7757:X 7754:[ 7748:E 7745:+ 7742:] 7739:X 7736:[ 7723:2 7719:] 7715:X 7712:[ 7706:E 7700:( 7690:= 7686:) 7680:] 7675:2 7671:X 7667:[ 7661:E 7654:2 7650:] 7646:X 7643:[ 7637:E 7631:( 7621:= 7596:σ 7590:μ 7568:. 7563:1 7553:2 7544:e 7538:= 7535:] 7532:X 7529:[ 7496:] 7493:X 7490:[ 7484:E 7479:] 7476:X 7473:[ 7443:] 7440:X 7437:[ 7401:, 7396:1 7386:2 7377:e 7367:2 7356:2 7353:1 7347:+ 7340:e 7336:= 7331:1 7321:2 7312:e 7306:] 7303:X 7300:[ 7294:E 7291:= 7286:] 7283:X 7280:[ 7269:= 7262:] 7259:X 7256:[ 7243:, 7240:) 7237:1 7227:2 7218:e 7214:( 7207:2 7199:+ 7193:2 7189:e 7185:= 7182:) 7179:1 7169:2 7160:e 7156:( 7151:2 7147:) 7143:] 7140:X 7137:[ 7131:E 7128:( 7125:= 7120:2 7116:] 7112:X 7109:[ 7103:E 7097:] 7092:2 7088:X 7084:[ 7078:E 7075:= 7068:] 7065:X 7062:[ 7049:, 7042:2 7034:2 7031:+ 7025:2 7021:e 7017:= 7010:] 7005:2 7001:X 6997:[ 6991:E 6984:, 6977:2 6966:2 6963:1 6957:+ 6950:e 6946:= 6939:] 6936:X 6933:[ 6927:E 6909:X 6891:. 6884:2 6874:2 6870:n 6864:2 6861:1 6856:+ 6850:n 6846:e 6842:= 6839:] 6834:n 6830:X 6826:[ 6820:E 6806:X 6796:n 6790:n 6751:2 6741:2 6738:1 6729:e 6705:. 6700:] 6697:X 6694:[ 6680:] 6677:X 6674:[ 6665:= 6656:2 6647:e 6632:e 6628:= 6621:2 6611:2 6608:1 6603:+ 6596:e 6592:= 6589:] 6586:X 6583:[ 6577:E 6515:1 6503:e 6499:= 6496:] 6493:X 6490:[ 6456:2 6447:e 6443:= 6440:] 6437:X 6434:[ 6395:= 6386:e 6382:= 6379:] 6376:X 6373:[ 6333:= 6324:e 6320:= 6317:] 6314:X 6311:[ 6269:( 6247:y 6221:0 6213:x 6209:e 6182:y 6173:= 6170:x 6150:) 6147:0 6141:y 6132:( 6129:p 6106:= 6100:, 6097:1 6094:= 6071:y 6015:W 5989:) 5980:e 5974:2 5966:t 5963:i 5957:( 5954:W 5951:+ 5948:1 5942:) 5933:2 5925:2 5920:) 5911:e 5905:2 5897:t 5894:i 5888:( 5885:W 5882:2 5879:+ 5876:) 5867:e 5861:2 5853:t 5850:i 5844:( 5839:2 5835:W 5824:( 5808:) 5805:t 5802:( 5776:t 5756:) 5753:t 5750:( 5715:2 5711:/ 5705:2 5695:2 5691:n 5687:+ 5681:n 5677:e 5670:! 5667:n 5660:n 5656:) 5652:t 5649:i 5646:( 5633:0 5630:= 5627:n 5600:t 5594:t 5579:] 5574:X 5571:t 5568:i 5564:e 5560:[ 5554:E 5528:t 5508:] 5503:X 5500:t 5496:e 5492:[ 5486:E 5458:) 5453:2 5445:n 5442:+ 5436:( 5430:) 5427:x 5424:( 5411:= 5408:z 5383:2 5379:/ 5373:2 5363:2 5359:n 5355:+ 5349:n 5345:e 5341:= 5338:] 5333:n 5329:X 5325:[ 5319:E 5281:. 5278:) 5275:1 5265:j 5262:i 5253:e 5249:( 5244:) 5239:j 5236:j 5228:+ 5223:i 5220:i 5212:( 5207:2 5204:1 5199:+ 5194:j 5186:+ 5181:i 5172:e 5168:= 5163:j 5160:i 5156:] 5151:Y 5147:[ 5111:, 5104:i 5101:i 5091:2 5088:1 5083:+ 5078:i 5069:e 5065:= 5060:i 5056:] 5051:Y 5047:[ 5041:E 5017:Y 4995:X 4974:) 4969:i 4965:X 4961:( 4952:= 4947:i 4943:Y 4918:) 4909:, 4901:( 4896:N 4887:X 4844:) 4835:2 4819:x 4803:( 4791:2 4788:1 4783:= 4779:] 4774:) 4766:2 4750:x 4738:( 4728:+ 4725:1 4721:[ 4715:2 4712:1 4681:) 4678:1 4672:, 4669:0 4663:( 4655:N 4600:) 4585:) 4582:x 4573:( 4567:( 4560:= 4557:) 4554:x 4551:( 4546:X 4542:F 4502:. 4495:) 4486:2 4475:2 4465:2 4461:) 4451:x 4442:( 4429:( 4402:2 4391:x 4384:1 4379:= 4363:x 4350:1 4344:) 4326:x 4311:( 4301:= 4290:) 4272:x 4257:( 4250:x 4245:d 4238:d 4231:) 4216:x 4204:( 4194:= 4183:) 4165:x 4150:( 4136:x 4131:d 4124:d 4118:= 4106:] 4098:x 4086:X 4072:[ 4061:X 4055:P 4043:x 4038:d 4031:d 4025:= 4013:] 4005:x 3999:X 3991:[ 3980:X 3974:P 3962:x 3957:d 3950:d 3944:= 3937:) 3934:x 3931:( 3926:X 3922:f 3887:) 3881:1 3878:, 3875:0 3869:( 3864:N 3784:) 3779:2 3771:, 3765:( 3760:N 3752:) 3749:X 3746:( 3718:: 3710:2 3659:X 3632:) 3623:2 3615:, 3605:( 3592:X 3566:X 3452:e 3448:= 3406:e 3402:= 3367:. 3360:) 3352:2 3347:X 3333:2 3328:X 3314:+ 3311:1 3307:( 3297:= 3292:2 3260:) 3244:2 3239:X 3231:+ 3226:2 3221:X 3205:2 3200:X 3190:( 3180:= 3154:, 3146:2 3141:X 3111:X 3083:1 3077:a 3071:0 3051:, 3043:Y 3039:a 3010:Y 3006:e 2982:. 2976:1 2970:b 2967:, 2964:a 2938:) 2935:X 2932:( 2924:b 2893:) 2890:X 2887:( 2879:a 2842:Y 2838:e 2831:X 2805:Z 2799:+ 2793:= 2790:Y 2757:X 2669:Z 2663:+ 2656:e 2652:= 2649:X 2626:0 2553:Z 2527:) 2525:X 2519:X 2450:, 2448:) 2446:Y 2442:X 2436:Y 2428:Y 2423:) 2421:X 2417:Y 2411:X 2360:) 2357:) 2348:) 2345:p 2342:( 2337:1 2326:( 2317:1 2314:( 2308:p 2302:1 2298:1 2289:2 2284:2 2274:+ 2267:e 2263:= 2236:p 2228:) 2224:) 2221:1 2215:p 2212:2 2209:( 2201:1 2190:+ 2176:2 2161:( 2151:+ 2148:1 2135:2 2130:2 2120:+ 2113:e 2107:2 2104:1 2064:) 2057:1 2054:+ 2043:2 2039:] 2035:X 2032:[ 2024:E 2008:] 2005:X 2002:[ 1986:( 1974:= 1947:, 1940:) 1926:1 1923:+ 1912:2 1908:] 1904:X 1901:[ 1893:E 1877:] 1874:X 1871:[ 1846:] 1843:X 1840:[ 1832:E 1824:( 1814:= 1775:) 1761:2 1749:2 1742:0 1735:0 1722:2 1710:1 1702:( 1660:2 1656:/ 1650:2 1640:2 1636:n 1632:+ 1626:n 1619:e 1612:! 1609:n 1599:n 1595:) 1591:t 1585:i 1582:( 1566:0 1563:= 1560:n 1507:) 1495:2 1492:1 1486:+ 1479:e 1458:2 1449:( 1440:2 1401:6 1394:) 1388:2 1380:2 1376:( 1363:3 1360:+ 1356:) 1350:2 1339:3 1335:( 1322:2 1319:+ 1315:) 1309:2 1298:4 1294:( 1281:1 1247:1 1241:) 1236:2 1228:( 1216:] 1209:2 1206:+ 1202:) 1197:2 1189:( 1175:[ 1139:) 1133:2 1125:+ 1116:2 1112:( 1098:] 1091:1 1085:) 1080:2 1072:( 1059:[ 1023:) 1014:2 996:( 955:) 943:( 902:) 893:2 888:2 878:+ 868:( 830:) 827:) 824:p 821:( 816:1 802:+ 796:( 787:= 762:) 758:) 755:1 749:p 746:2 743:( 735:1 720:2 712:2 707:+ 700:( 661:) 646:) 643:x 640:( 628:( 621:= 617:] 612:) 601:2 582:x 567:( 557:+ 554:1 550:[ 544:2 537:1 499:) 490:2 482:2 476:2 471:) 458:x 448:( 437:( 410:2 402:x 395:1 359:) 350:+ 347:, 344:0 338:( 332:x 312:) 293:0 256:) 247:+ 244:, 232:( 191:) 182:2 173:, 163:( 123:0 120:= 20:)

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.