3141:). Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a degenerate multivariate normal approximation to the multinomial distribution (the numbers in each category add up to the total sample size, which is considered fixed). Pearson showed that the chi-squared distribution arose from such a multivariate normal approximation to the multinomial distribution, taking careful account of the statistical dependence (negative correlations) between numbers of observations in different categories.
14184:
2262:. The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-squared distribution have low probability.
7675:
2058:) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used.
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53:
41:
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856:
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14624:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 940.
3628:
5569:
2289:
Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and
Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable
14551:
15242:
Table B.2. Dr. Jacqueline S. McLaughlin at The
Pennsylvania State University. In turn citing: R. A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed., Table IV. Two values have been corrected, 7.82 with 7.81 and 4.60 with
4870:
11261:
715:
5340:
10064:
2277:
distribution rather than the normal approximation or the chi-squared approximation for a small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for a small sample size, and it is preferable to use
5898:
10507:
442:
13395:
1952:
The chi-squared distribution is used primarily in hypothesis testing, and to a lesser extent for confidence intervals for population variance when the underlying distribution is normal. Unlike more widely known distributions such as the
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12921:
12019:
independent, zero-mean, unit-variance
Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.
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13291:
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1604:
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7199:
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12366:
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12999:
7275:
1338:
1095:
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Pearson, Karl (1914). "On the probability that two independent distributions of frequency are really samples of the same population, with special reference to recent work on the identity of
Trypanosome strains".
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349:
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544:
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939:
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2030:
The primary reason for which the chi-squared distribution is extensively used in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the
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8626:
8359:
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11442:{\displaystyle {\frac {1}{\left({\frac {w_{1}}{X_{1}}},\ldots ,{\frac {w_{p}}{X_{p}}}\right)\Sigma \left({\frac {w_{1}}{X_{1}}},\ldots ,{\frac {w_{p}}{X_{p}}}\right)^{\top }}}\sim \chi _{1}^{2}.}
9619:
9480:
2273:) and this leads also to optimality properties of generalised LRTs. However, the normal and chi-squared approximations are only valid asymptotically. For this reason, it is preferable to use the
851:{\displaystyle {\begin{aligned}{\frac {k}{2}}&+\log \left(2\Gamma {\Bigl (}{\frac {k}{2}}{\Bigr )}\right)\\&\!+\left(1-{\frac {k}{2}}\right)\psi \left({\frac {k}{2}}\right)\end{aligned}}}
9710:
720:
4227:
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4105:
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1817:
1661:
10766:
9970:
9924:
7104:
12200:
11832:
11780:
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11673:
6400:
11253:
5216:
9760:
9669:
9285:
6120:
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2260:
1428:
9108:
1532:
6717:
179:
9800:
11193:
5753:
11532:
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8814:
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12803:
12563:
12467:
12303:
12072:
10394:
9975:
9330:
9248:
7686:) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.
5799:
5701:
3807:
1276:
230:
11081:
8173:
8064:
7871:
7804:
1476:
8250:
6057:{\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\sim \operatorname {Gamma} \left(\alpha =n\,k/2,\theta =2/n\right)\qquad {\text{where }}X_{i}\sim \chi ^{2}(k)}
8285:
7355:
7326:
12758:
7436:
6355:
5100:
4315:
702:
667:
130:
8124:
7984:
1003:
97:
8953:
7910:
1230:
594:
8458:
8212:
8091:
3666:
3029:
2160:
14271:
12004:
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4020:
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8656:
4924:
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13605:
11040:
10893:
8011:
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2912:
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1144:
9135:
6648:
3916:
14420:{\displaystyle f(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}
11014:
10965:
7758:
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5870:
2193:
1378:
1358:
362:
13573:, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and non-significant results.
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in papers of 1875–6, where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the
12811:
12761:
10653:
7942:
6328:
5829:
5581:
It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if
3475:
3334:{\displaystyle f(x;\,k)={\begin{cases}{\dfrac {x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma \left({\frac {k}{2}}\right)}},&x>0;\\0,&{\text{otherwise}}.\end{cases}}}
626:
12659:
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11121:
11101:
10867:
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10824:
10786:
10693:
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10633:
10613:
10586:
10566:
7824:
7732:
7712:
7395:
7375:
6838:
6679:
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6284:
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5147:
5022:
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3049:
2986:
2408:
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2213:
2119:
2099:
2079:
1189:
1166:
470:
1961:, the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others:
3708:
13541:
7012:{\displaystyle \operatorname {E} (X^{m})=k(k+2)(k+4)\cdots (k+2m-2)=2^{m}{\frac {\Gamma \left(m+{\frac {k}{2}}\right)}{\Gamma \left({\frac {k}{2}}\right)}}.}
6171:
13408:
13119:
15533:
10236:
6611:{\displaystyle h=\int _{0}^{\infty }f(x;\,k)\ln f(x;\,k)\,dx={\frac {k}{2}}+\ln \left+\left(1-{\frac {k}{2}}\right)\,\psi \!\left({\frac {k}{2}}\right),}
15165:
Bausch, J. (2013). "On the
Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua".
1832:
14172:). The idea of a family of "chi-squared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.
12207:
8963:
3121:
In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large
3623:{\displaystyle F(x;\,k)={\frac {\gamma ({\frac {k}{2}},\,{\frac {x}{2}})}{\Gamma ({\frac {k}{2}})}}=P\left({\frac {k}{2}},\,{\frac {x}{2}}\right),}
13219:
12229:
The noncentral chi-squared distribution is obtained from the sum of the squares of independent
Gaussian random variables having unit variance and
5564:{\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}~=~Z^{\top }\!MZ~=~X^{\top }\!Q^{\top }\!MQX~=~X_{2}^{2}+...+X_{n}^{2}~\sim ~\chi _{n-1}^{2},}
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2269:(LRT). LRTs have several desirable properties; in particular, simple LRTs commonly provide the highest power to reject the null hypothesis (
1281:
1016:
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945:
243:
4949:
2265:
An additional reason that the chi-squared distribution is widely used is that it turns up as the large sample distribution of generalized
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6804:
are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the
483:
16888:
16145:
10184:
4237:
3432:
1992:
869:
16053:
14546:{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}
8398:
4428:
16840:
15253:
14576:
14145:, pp. xxxi–xxxiii, 26–28, Table XII). The name "chi-square" ultimately derives from Pearson's shorthand for the exponent in a
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Following are some of the most common situations in which the chi-squared distribution arises from a
Gaussian-distributed sample.
11537:
6722:
16706:
15918:
15677:
15526:
14980:
Bartlett, M. S.; Kendall, D. G. (1946). "The
Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation".
14203:
12681:. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a
12077:
7678:
Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (
3386:
12371:
8301:
16601:
16365:
9492:
4865:{\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}~=~\sum _{t=1}^{n}Z_{t}^{2}-n{\bar {Z}}^{2}~=~Z^{\top }Z~=:~Z^{\top }\!MZ}
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of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as
16360:
16304:
16202:
15964:
15602:
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Ueber die
Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen
14213:
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12203:
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The chi-squared distribution is also naturally related to other distributions arising from the
Gaussian. In particular,
16878:
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4141:
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10119:
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9335:
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7114:
The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart bounds are:
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Tables of the chi-squared cumulative distribution function are widely available and the function is included in many
3815:
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Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:
5335:{\displaystyle X:=Q^{\top }\!Z\sim {\mathcal {N}}({\bar {0}},Q^{\top }\!1\!\!1Q)={\mathcal {N}}({\bar {0}},1\!\!1)}
3482:
3448:
1757:
1609:
355:
10698:
9929:
9883:
7041:
16485:
16398:
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16279:
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15883:
15848:
12206:. A closed expression for this distribution is not known. It may be, however, approximated efficiently using the
12141:
11785:
11733:
11678:
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10361:
9065:
6360:
3669:
1932:
1168:
1009:
15502:
Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator
11198:
16499:
16416:
16253:
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16000:
15878:
15853:
15717:
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14233:
9719:
9628:
9253:
8402:
6093:
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2218:
1383:
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9074:
1489:
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16681:
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15682:
15624:
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15441:
14228:
14223:
14134:
6684:
1974:
1965:
143:
15489:
10059:{\displaystyle {\tfrac {X}{X+Y}}\sim \operatorname {Beta} ({\tfrac {\nu _{1}}{2}},{\tfrac {\nu _{2}}{2}})\,}
9767:
7714:
independent random variables with finite mean and variance, it converges to a normal distribution for large
16909:
16651:
16641:
16332:
16258:
15959:
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11146:
10589:
10532:
5706:
3946:
3150:
236:
16711:
14686:
Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "Chi-Square Distributions including Chi and Rayleigh".
11482:
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15597:
15436:
14554:
13401:
13099:
independent random variables that have probability distributions related to the chi-squared distribution:
12766:
12686:
12530:
12434:
12270:
12031:
10523:
10517:
10367:
9297:
9215:
5758:
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2010:
1243:
193:
11045:
8136:
8027:
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1439:
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16207:
16160:
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16005:
15985:
15828:
15702:
15582:
15296:
13492:
8217:
8014:
862:
19:
This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see
8255:
7331:
7283:
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12711:
10792:
10541:
7400:
6333:
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4268:
680:
639:
102:
15269:
15267:
10502:{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |^{\beta }}{\alpha }}\sim \chi _{2n/\beta }^{2}\,}
8096:
7950:
3190:
952:
65:
16521:
16309:
16075:
16034:
15949:
15843:
15808:
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15592:
15542:
14669:
14115:
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12517:
12428:
9379:
8926:
7883:
2270:
1958:
1702:
1667:
1203:
557:
15217:
den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review",
8437:
8178:
8069:
3636:
2991:
2124:
16820:
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16130:
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15888:
15838:
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15614:
15264:
14244:
11969:
8391:
6090:
going to infinity, a Gamma distribution converges towards a normal distribution with expectation
3993:
3955:
2526:
15990:
8635:
4900:
4590:
437:{\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)\;}
16686:
16674:
16663:
16545:
16441:
16248:
15692:
15672:
15577:
15403:
13618:
13583:
13390:{\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}
11019:
10872:
7989:
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2890:
2576:
2279:
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1122:
14956:
14949:
9113:
6624:
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3347:
16810:
16767:
16611:
16286:
16140:
16120:
16017:
15587:
14619:
12916:{\displaystyle \sum _{i=1}^{n}(X_{i}-{\overline {X_{i}}})^{2}\sim \sigma ^{2}\chi _{n-1}^{2}}
10999:
10937:
10903:
9622:
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1980:
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15479:
Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history
15184:
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15020:
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12690:
11942:
11915:
10336:
9875:
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2944:
2620:
2413:
1483:
1237:
16065:
15493:
demonstration showing the chi-squared sampling distribution of various statistics, e. g. Σ
11042:
positive-semidefinite covariance matrix with strictly positive diagonal entries, then for
10638:
7919:
6309:
3454:
607:
8:
16794:
16319:
16299:
16269:
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16125:
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15483:
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12608:
12524:
10970:
10908:
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10512:
9483:
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8361:
by back-transforming from the mean, which is also the median, of the normal distribution.
7761:
5027:
3717:
1954:
1679:
708:
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9206:
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6823:
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6289:
6269:
6253:{\displaystyle {\overline {X}}\xrightarrow {n\to \infty } N(\mu =k,\sigma ^{2}=2\,k/n)}
5835:
5812:
5132:
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4929:
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3414:
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2198:
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2084:
2064:
1969:
1710:
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1479:
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1174:
1151:
1103:
455:
15506:
13479:{\displaystyle {\sqrt {\sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}
2081:
is a random variable sampled from the standard normal distribution, where the mean is
16723:
16150:
15893:
15823:
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15737:
15389:
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15351:
15204:
15108:
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14607:
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14070:
13201:{\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}
12682:
10899:
10067:
550:
15332:"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"
15151:
15051:
14842:
14073:(also known as "inverse CDF" or "ICDF") of the chi-squared distribution; e. g., the
15898:
15572:
15511:
15459:
15419:
15343:
15192:
15139:
15127:
15098:
15088:
15039:
14989:
14911:
14830:
14770:
14198:
13526:
13297:
12674:
10895:
10324:{\displaystyle \sum _{i=1}^{n}{\frac {2|X_{i}-\mu |}{\beta }}\sim \chi _{2n}^{2}\,}
9288:
6651:
1694:
20:
15347:
10799:
have mean zero yields a generalization of the chi-squared distribution called the
15239:
15232:
14815:
14643:
14621:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
14615:
14130:
12805:
12698:
7913:
4322:
1917:{\displaystyle Q\ \sim \ \chi ^{2}(k)\ \ {\text{or}}\ \ Q\ \sim \ \chi _{k}^{2}.}
1748:
1698:
1194:
673:
15222:
9046:{\displaystyle \|{\boldsymbol {N}}_{i=1,\ldots ,k}(0,1)\|^{2}\sim \chi _{k}^{2}}
3882:
directly. The integer recurrence of the gamma function makes it easy to compute
15971:
15143:
14761:
Ramsey, PH (1988). "Evaluating the Normal Approximation to the Binomial Test".
14114:
This distribution was first described by the German geodesist and statistician
12694:
12527:
is also a special case of the gamma distribution and thus we also have that if
11476:
10932:
3451:
and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (
3382:
2017:
448:
16903:
16594:
16342:
15629:
15463:
15423:
15355:
15313:
15285:
14925:
14916:
14125:
The distribution was independently rediscovered by the English mathematician
12247:
The generalized chi-squared distribution is obtained from the quadratic form
4233:
2283:
1986:
1714:
1706:
14899:
13286:{\displaystyle \sum _{i=1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}
3990:
on the lower and upper tails of the CDF may be obtained. For the cases when
15128:"Fast Randomization for Distributed Low-Bitrate Coding of Speech and Audio"
15112:
15093:
14611:
14150:
14126:
9860:{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Inv-} \chi _{\nu }^{2}\,}
9061:
standard normally distributed variables is a chi-squared distribution with
8549:{\displaystyle (\chi _{k}^{2}-k)/{\sqrt {2k}}~{\xrightarrow {d}}\ N(0,1)\,}
8127:
7674:
7032:
2875:{\displaystyle \chi ^{2}=\sum _{i=1}^{n}{\frac {(O_{i}-E_{i})^{2}}{E_{i}}}}
2773:
1599:{\displaystyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2s^{2})}
15018:
Pillai, Natesh S. (2016). "An unexpected encounter with Cauchy and Lévy".
12255:
is a zero-mean Gaussian vector having an arbitrary covariance matrix, and
15381:
7194:{\displaystyle \operatorname {P} (X-k\geq 2{\sqrt {kx}}+2x)\leq \exp(-x)}
4418:{\displaystyle \sum _{t=1}^{n}(Z_{t}-{\bar {Z}})^{2}\sim \chi _{n-1}^{2}}
3942:
2914:= Pearson's cumulative test statistic, which asymptotically approaches a
2032:
24:
14690:. Vol. 1 (Second ed.). John Wiley and Sons. pp. 415–493.
12697:, which is the distribution of the ratio of two independent chi-squared
6820:
The noncentral moments (raw moments) of a chi-squared distribution with
15501:
15043:
15001:
14782:
13005:
12670:
12431:
is also a special case of the gamma distribution, we also have that if
12361:{\displaystyle X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right)}
8405: in this section. Unsourced material may be challenged and removed.
7876:
to a standard normal distribution. However, convergence is slow as the
7377:
grows, the squared length of the vector is concentrated tightly around
2762:{\displaystyle \chi ^{2}={(m-Np)^{2} \over Np}+{(N-m-Nq)^{2} \over Nq}}
2003:
1709:, and in finding the confidence interval for estimating the population
1107:
14834:
12994:{\displaystyle {\overline {X_{i}}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}
12669:
The chi-squared distribution has numerous applications in inferential
12015:
The chi-squared distribution is obtained as the sum of the squares of
7270:{\displaystyle \operatorname {P} (k-X\geq 2{\sqrt {kx}})\leq \exp(-x)}
6262:
Note that we would have obtained the same result invoking instead the
1333:{\displaystyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2)}
1090:{\displaystyle (1-2\ln t)^{-k/2}{\text{ for }}0<t<{\sqrt {e}}\;}
14865:
12509:{\displaystyle X\sim \operatorname {Exp} \left({\frac {1}{2}}\right)}
10544:
can be obtained from normal distribution and chi-squared distribution
7986:
converges to normality much faster than the sampling distribution of
1998:
344:{\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\;}
15478:
15407:
14993:
14774:
14655:
10535:
can be obtained as a transformation of chi-squared distribution and
8517:
8380:
6189:
4997:{\displaystyle b_{1}:={\textstyle {\frac {1}{\sqrt {n}}}}{\bar {1}}}
15317:
15034:
13092:{\displaystyle X_{i}\sim N(\mu _{i},\sigma _{i}^{2}),i=1,\ldots ,k}
12678:
7877:
7028:
6808:. For derivation from more basic principles, see the derivation in
1927:
The chi-squared distribution has one parameter: a positive integer
632:
600:
15179:
14900:"Adaptive estimation of a quadratic functional by model selection"
12420:
using the scale parameterization of the gamma distribution) where
4022:(which include all of the cases when this CDF is less than half):
539:{\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}
15408:"Tables for Testing the Goodness of Fit of Theory to Observation"
14137:, published in 1900, with computed table of values published in (
14100:
13549:
as extreme in a chi-squared distribution. Accordingly, since the
13523:
10226:{\displaystyle X_{i}\sim \operatorname {Laplace} (\mu ,\beta )\,}
3443:
1977:
of goodness of fit of observed data to hypothetical distributions
52:
40:
14639:
14577:"Characteristic function of the central chi-square distribution"
12262:
934:{\displaystyle (1-2t)^{-k/2}{\text{ for }}t<{\frac {1}{2}}\;}
15276:, pp. 633–692, 27. Sampling Distributions under Normality.
15132:
IEEE/ACM Transactions on Audio, Speech, and Language Processing
14816:"An Elementary Proof of a Theorem of Johnson and Lindenstrauss"
14711:
Mood, Alexander; Graybill, Franklin A.; Boes, Duane C. (1974).
10846:
476:
16:
Probability distribution and special case of gamma distribution
12368:
using the rate parameterization of the gamma distribution (or
4494:{\displaystyle {\bar {Z}}={\frac {1}{n}}\sum _{t=1}^{n}Z_{t}.}
1822:
is distributed according to the chi-squared distribution with
14879:
Probability Distributions Involving Gaussian Random Variables
15330:
Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021).
13561:
than this point, subtracting the CDF value from 1 gives the
11616:{\displaystyle Y={\frac {{X_{1}}/{k_{1}}}{{X_{2}}/{k_{2}}}}}
6797:{\displaystyle \operatorname {E} (\ln(X))=\psi (k/2)+\ln(2)}
5852:
is distributed according to a gamma distribution with shape
4236:
for the CDF modeled after the cube of a Gaussian, see under
3051:, asserted by the null hypothesis that the fraction of type
1666:
The chi-squared distribution is one of the most widely used
12131:{\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {R} _{>0}}
10568:
degrees of freedom is defined as the sum of the squares of
9874:
The chi-squared distribution is a special case of type III
7035:
expansion of the logarithm of the characteristic function:
3327:
14982:
Supplement to the Journal of the Royal Statistical Society
14670:
Engineering Statistics Handbook – Chi-Squared Distribution
13491:
The chi-squared distribution is also often encountered in
12413:{\displaystyle X\sim \Gamma \left({\frac {k}{2}},2\right)}
8354:{\displaystyle k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}
2050:
of the test statistic approaches the normal distribution (
14814:
Dasgupta, Sanjoy D. A.; Gupta, Anupam K. (January 2003).
9526:{\displaystyle X\sim \operatorname {Erlang} (k,\lambda )}
8298:
leads directly to the commonly used median approximation
7599:{\displaystyle Pr(\|v\|^{2}\in )\geq 1-e^{-n^{\alpha }}}
3411:
For derivations of the pdf in the cases of one, two and
15318:
Earliest Known Uses of Some of the Words of Mathematics
14685:
13569:-value, below the chosen significance level, indicates
12701:, each divided by their respective degrees of freedom.
12236:
11905:{\displaystyle X_{1}+X_{2}\sim \chi _{k_{1}+k_{2}}^{2}}
4587:
independent normally distributed random variables, and
4560:{\displaystyle Z\sim {\mathcal {N}}({\bar {0}},1\!\!1)}
2286:
is always more powerful than the normal approximation.
2042:-test. For these hypothesis tests, as the sample size,
2027:-tests, analysis of variance, and regression analysis.
15386:
A history of mathematical statistics from 1750 to 1930
14946:
14480:
13529:
12218:
10034:
10012:
9980:
9810:
8770:{\displaystyle X=\lim _{\nu _{2}\to \infty }\nu _{1}Y}
6810:
moment-generating function of the sufficient statistic
6266:, noting that for each chi-squared variable of degree
4967:
4786:
4259:
The following is a special case of Cochran's theorem.
3678:
3350:
1701:
of an observed distribution to a theoretical one, the
14433:
14279:
14247:
13621:
13586:
13411:
13307:
13222:
13122:
13014:
12929:
12814:
12769:
12714:
12639:
12611:
12591:
12571:
12533:
12475:
12437:
12374:
12315:
12273:
12144:
12080:
12034:
11972:
11945:
11918:
11840:
11788:
11736:
11681:
11629:
11540:
11485:
11461:
11264:
11201:
11149:
11129:
11109:
11089:
11048:
11022:
11002:
10973:
10940:
10911:
10875:
10855:
10832:
10812:
10774:
10701:
10681:
10661:
10641:
10635:-dimensional Gaussian random vector with mean vector
10621:
10601:
10574:
10554:
10402:
10370:
10339:
10239:
10187:
10128:
10079:
9978:
9932:
9886:
9808:
9770:
9722:
9679:
9631:
9588:
9539:
9495:
9436:
9395:
9338:
9300:
9256:
9218:
9198:{\displaystyle cX\sim \Gamma (k=\nu /2,\theta =2c)\,}
9143:
9116:
9077:
8966:
8929:
8879:
8828:
8783:
8724:
8711:{\displaystyle Y\sim \mathrm {F} (\nu _{1},\nu _{2})}
8667:
8638:
8621:{\displaystyle \chi _{k}^{2}\sim {\chi '}_{k}^{2}(0)}
8568:
8466:
8440:
8304:
8258:
8220:
8181:
8139:
8099:
8072:
8030:
7992:
7953:
7922:
7886:
7832:
7812:
7770:
7740:
7720:
7700:
7694:, because the chi-squared distribution is the sum of
7632:
7612:
7444:
7403:
7383:
7363:
7334:
7286:
7208:
7123:
7044:
6849:
6826:
6725:
6687:
6667:
6627:
6423:
6363:
6336:
6312:
6292:
6272:
6174:
6128:
6096:
6076:
5901:
5878:
5858:
5838:
5815:
5761:
5709:
5668:
5641:
5587:
5348:
5219:
5155:
5135:
5108:
5056:
5030:
5010:
4952:
4932:
4903:
4878:
4622:
4593:
4573:
4514:
4431:
4331:
4271:
4144:
4115:
4028:
3996:
3958:
3924:
3888:
3818:
3749:
3720:
3639:
3494:
3457:
3417:
3393:
3194:
3162:
3127:
3104:
3077:
3057:
3037:
2994:
2974:
2947:
2920:
2893:
2785:
2658:
2623:
2579:
2529:
2454:
2416:
2396:
2376:
2356:
2299:
2221:
2201:
2168:
2127:
2107:
2087:
2067:
1835:
1760:
1612:
1540:
1492:
1442:
1386:
1366:
1346:
1284:
1246:
1206:
1177:
1154:
1125:
1019:
955:
872:
718:
683:
642:
610:
560:
486:
458:
365:
246:
196:
146:
105:
68:
15541:
15125:
14179:
9614:{\displaystyle X\sim \operatorname {Rayleigh} (1)\,}
9475:{\displaystyle X\sim \operatorname {Erlang} (k,1/2)}
15484:
Course notes on Chi-Squared Goodness of Fit Testing
9705:{\displaystyle X\sim \operatorname {Maxwell} (1)\,}
7764:, so the difference is ignorable. Specifically, if
3438:
1693:The chi-squared distribution is used in the common
1380:the scale parameter of the gamma distribution) and
14948:
14545:
14419:
14265:
13634:
13599:
13535:
13478:
13389:
13285:
13200:
13091:
12993:
12915:
12797:
12752:
12653:
12625:
12597:
12577:
12557:
12508:
12461:
12412:
12360:
12297:
12194:
12130:
12066:
11998:
11958:
11931:
11904:
11826:
11774:
11719:
11667:
11615:
11526:
11467:
11441:
11247:
11187:
11135:
11115:
11095:
11075:
11034:
11008:
10985:
10959:
10923:
10887:
10861:
10838:
10818:
10780:
10760:
10687:
10667:
10647:
10627:
10607:
10580:
10560:
10526:can be obtained from chi-squared distribution and
10501:
10388:
10352:
10323:
10225:
10171:
10110:
10058:
9964:
9918:
9859:
9794:
9754:
9704:
9663:
9613:
9572:
9525:
9474:
9422:
9370:
9324:
9279:
9242:
9197:
9129:
9102:
9045:
8947:
8915:
8865:
8808:
8769:
8710:
8650:
8620:
8548:
8452:
8353:
8279:
8244:
8206:
8167:
8118:
8085:
8058:
8005:
7978:
7936:
7904:
7865:
7818:
7798:
7752:
7726:
7706:
7658:
7618:
7598:
7430:
7389:
7369:
7349:
7320:
7269:
7193:
7098:
7011:
6832:
6796:
6711:
6673:
6642:
6610:
6394:
6349:
6322:
6298:
6278:
6252:
6158:
6114:
6082:
6056:
5884:
5864:
5844:
5821:
5793:
5747:
5695:
5654:
5627:
5563:
5334:
5205:
5141:
5121:
5094:
5042:
5016:
4996:
4938:
4918:
4889:
4864:
4608:
4579:
4559:
4493:
4417:
4309:
4221:
4127:
4099:
4014:
3978:
3930:
3910:
3874:
3801:
3732:
3699:
3660:
3622:
3469:
3423:
3399:
3373:
3333:
3133:
3110:
3090:
3063:
3043:
3023:
2980:
2960:
2933:
2906:
2874:
2761:
2641:
2609:
2565:
2512:
2434:
2402:
2382:
2362:
2339:
2254:
2207:
2187:
2154:
2113:
2093:
2073:
1916:
1811:
1655:
1598:
1526:
1470:
1422:
1372:
1352:
1332:
1270:
1224:
1183:
1160:
1138:
1089:
997:
933:
850:
696:
661:
620:
588:
538:
464:
436:
343:
224:
173:
124:
91:
15336:Communications in Statistics - Theory and Methods
15323:
14715:(Third ed.). McGraw-Hill. pp. 241–246.
13557:gives the probability of having obtained a value
13545:is the probability of observing a test statistic
13503:
8339:
8310:
6586:
5464:
5453:
5427:
5325:
5324:
5283:
5282:
5278:
5236:
4883:
4882:
4855:
4778:
4777:
4550:
4549:
4222:{\displaystyle 1-F(zk;\,k)\leq (ze^{1-z})^{k/2}.}
1751:random variables, then the sum of their squares,
793:
776:
759:
524:
495:
121:
16901:
14710:
14606:
10511:chi-squared distribution is a transformation of
8916:{\displaystyle X=\lim _{\nu _{2}\to \infty }Y\,}
8887:
8866:{\displaystyle Y\sim \mathrm {F} (1,\nu _{2})\,}
8732:
8214:is approximately normally distributed with mean
8093:is approximately normally distributed with mean
6159:{\displaystyle \sigma ^{2}=\alpha \,\theta ^{2}}
2772:The expression on the right is of the form that
2513:{\displaystyle \chi ^{2}={(m-Np)^{2} \over Npq}}
2340:{\displaystyle \chi ={m-Np \over {\sqrt {Npq}}}}
2009:It is also a component of the definition of the
561:
14979:
14794:
14792:
14681:
14679:
14677:
10520:is a transformation of chi-squared distribution
10111:{\displaystyle X\sim \operatorname {U} (0,1)\,}
9371:{\displaystyle X\sim \operatorname {Exp} (1/2)}
4100:{\displaystyle F(zk;\,k)\leq (ze^{1-z})^{k/2}.}
3875:{\displaystyle f(x;\,2)={\frac {1}{2}}e^{-x/2}}
3144:
3031:= the expected (theoretical) frequency of type
1826:degrees of freedom. This is usually denoted as
1171:is the distribution of a sum of the squares of
16915:Infinitely divisible probability distributions
15064:
15013:
15011:
14897:
14069:These values can be calculated evaluating the
12664:
1935:(the number of random variables being summed,
15527:
14813:
13553:(CDF) for the appropriate degrees of freedom
12263:Gamma, exponential, and related distributions
10172:{\displaystyle -2\log(X)\sim \chi _{2}^{2}\,}
9573:{\displaystyle 2\lambda X\sim \chi _{2k}^{2}}
2215:is an example of a chi-squared distribution:
1812:{\displaystyle Q\ =\sum _{i=1}^{k}Z_{i}^{2},}
1656:{\displaystyle X\sim {\text{W}}_{1}(s^{2},k)}
15254:"Chi-squared Distribution | R Tutorial"
14789:
14674:
14157:for what would appear in modern notation as
12213:
10761:{\displaystyle X=(Y-\mu )^{T}C^{-1}(Y-\mu )}
9965:{\displaystyle Y\sim \chi _{\nu _{2}}^{2}\,}
9919:{\displaystyle X\sim \chi _{\nu _{1}}^{2}\,}
9016:
8967:
7760:the distribution is sufficiently close to a
7461:
7454:
7099:{\displaystyle \kappa _{n}=2^{n-1}(n-1)!\,k}
2390:trials, where the probability of success is
1682:. This distribution is sometimes called the
15126:Bäckström, T.; Fischer, J. (January 2018).
15008:
14729:
12605:is Erlang distributed with shape parameter
12195:{\displaystyle X=\sum _{i=1}^{n}a_{i}X_{i}}
11827:{\displaystyle X_{2}\sim \chi _{k_{2}}^{2}}
11775:{\displaystyle X_{1}\sim \chi _{k_{1}}^{2}}
11720:{\displaystyle X_{2}\sim \chi _{k_{2}}^{2}}
11668:{\displaystyle X_{1}\sim \chi _{k_{1}}^{2}}
6395:{\displaystyle \sigma ^{2}={\frac {2k}{n}}}
6330:(and hence the variance of the sample mean
5635:are independent chi-squared variables with
3812:which can be easily derived by integrating
15534:
15520:
15329:
15312:, International Statistical Review, 1983,
14738:Understanding Advanced Statistical Methods
14122:("Helmertian") or "Helmert distribution".
11248:{\displaystyle w_{i}\geq 0,i=1,\ldots ,p,}
10795:unit-variance Gaussian variables which do
8350:
6067:
3433:Proofs related to chi-squared distribution
2445:Squaring both sides of the equation gives
2195:. The distribution of the random variable
1086:
930:
617:
585:
535:
433:
395:
340:
294:
221:
88:
15178:
15102:
15092:
15033:
14915:
14798:
14754:
14600:
12115:
10498:
10320:
10222:
10168:
10107:
10055:
9961:
9915:
9856:
9755:{\displaystyle X^{2}\sim \chi _{3}^{2}\,}
9751:
9701:
9664:{\displaystyle X^{2}\sim \chi _{2}^{2}\,}
9660:
9610:
9280:{\displaystyle {\sqrt {X}}\sim \chi _{k}}
9194:
9126:
9099:
8912:
8862:
8545:
8421:Learn how and when to remove this message
7337:
7092:
6582:
6526:
6489:
6482:
6457:
6316:
6235:
6145:
6115:{\displaystyle \mu =\alpha \cdot \theta }
5979:
5628:{\displaystyle X_{i},i={\overline {1,n}}}
4166:
4044:
3901:
3831:
3762:
3601:
3539:
3507:
3175:
3153:(pdf) of the chi-squared distribution is
2255:{\displaystyle \ Q\ \sim \ \chi _{1}^{2}}
1423:{\displaystyle X\sim {\text{W}}_{1}(1,k)}
658:
417:
155:
15402:
14881:, New York: Springer, 2002, eq. (2.35),
14735:
14713:Introduction to the Theory of Statistics
14574:
14138:
13498:
9103:{\displaystyle X\sim \chi _{\nu }^{2}\,}
8370:
7673:
7669:
6659:maximum entropy probability distribution
3442:
1715:Friedman's analysis of variance by ranks
1527:{\displaystyle X\sim s^{2}\chi _{k}^{2}}
15448:
15065:Wilson, E. B.; Hilferty, M. M. (1931).
14898:Laurent, B.; Massart, P. (2000-10-01).
14204:Scaled inverse chi-squared distribution
14142:
8972:
8291:, see (18.24), p. 426 of Johnson.
7826:tends to infinity, the distribution of
6712:{\displaystyle \operatorname {E} (X)=k}
5703:degrees of freedom, respectively, then
2370:is the observed number of successes in
2054:). Because the test statistic (such as
174:{\displaystyle k\in \mathbb {N} ^{*}~~}
16902:
15507:Values of the Chi-squared distribution
15164:
15017:
14760:
9795:{\displaystyle X\sim \chi _{\nu }^{2}}
8130:, see (18.23), p. 426 of Johnson.
6806:Expectation of the log moment of gamma
15515:
15486:from Yale University Stats 101 class.
15310:Karl Pearson and the Chi-Squared Test
15294:Zeitschrift für Mathematik und Physik
14092:as in the table above, noticing that
13607:for the first 10 degrees of freedom.
12023:
11188:{\displaystyle w_{1}+\cdots +w_{p}=1}
10849:standard normal random variables and
6815:
6166:, the sample mean converges towards:
5748:{\displaystyle Y=X_{1}+\cdots +X_{n}}
2968:= the number of observations of type
1705:of two criteria of classification of
1686:, a special case of the more general
16884:
15380:
15273:
14214:Generalized chi-squared distribution
12243:Generalized chi-squared distribution
12237:Generalized chi-squared distribution
12208:property of characteristic functions
12204:Generalized Chi-squared Distribution
12074:are chi square random variables and
11834:are statistically independent, then
11527:{\displaystyle Y\sim F(k_{1},k_{2})}
9423:{\displaystyle X\sim \chi _{2k}^{2}}
8809:{\displaystyle \chi _{\nu _{1}}^{2}}
8403:adding citations to reliable sources
8374:
6657:The chi-squared distribution is the
5213:is an orthogonal matrix. Since also
5206:{\displaystyle Q:=(b_{1},...,b_{n})}
4248:
3118:= the number of cells in the table.
2649:, this equation can be rewritten as
2162:. Now, consider the random variable
14688:Continuous Univariate Distributions
14219:Noncentral chi-squared distribution
13213:noncentral chi-squared distribution
12798:{\displaystyle N(\mu ,\sigma ^{2})}
12558:{\displaystyle X\sim \chi _{k}^{2}}
12462:{\displaystyle X\sim \chi _{2}^{2}}
12298:{\displaystyle X\sim \chi _{k}^{2}}
12225:Noncentral chi-squared distribution
12219:Noncentral chi-squared distribution
12067:{\displaystyle X_{1},\ldots ,X_{n}}
10801:noncentral chi-squared distribution
10537:Noncentral chi-squared distribution
10389:{\displaystyle \mu ,\alpha ,\beta }
9325:{\displaystyle X\sim \chi _{2}^{2}}
9243:{\displaystyle X\sim \chi _{k}^{2}}
8630:noncentral chi-squared distribution
7734:. For many practical purposes, for
6070:, given that for a shape parameter
5794:{\displaystyle k_{1}+\cdots +k_{n}}
5696:{\displaystyle i={\overline {1,n}}}
4238:Noncentral chi-squared distribution
3802:{\displaystyle F(x;\,2)=1-e^{-x/2}}
3740:this function has the simple form:
1688:noncentral chi-squared distribution
1271:{\displaystyle X\sim \chi _{k}^{2}}
225:{\displaystyle x\in [0,+\infty )\;}
13:
15373:
14465:
14434:
14374:
14257:
13576:The table below gives a number of
12381:
12322:
12010:
11410:
11338:
11076:{\displaystyle X\sim N(0,\Sigma )}
11067:
11003:
10086:
9831:
9828:
9825:
9153:
8904:
8836:
8749:
8675:
8447:
8168:{\displaystyle X\sim \chi ^{2}(k)}
8059:{\displaystyle X\sim \chi ^{2}(k)}
7866:{\displaystyle (X-k)/{\sqrt {2k}}}
7799:{\displaystyle X\sim \chi ^{2}(k)}
7209:
7124:
6982:
6951:
6850:
6726:
6688:
6527:
6440:
6196:
5459:
5448:
5422:
5298:
5273:
5245:
5231:
4850:
4825:
4766:
4523:
4321:identically distributed (i.i.d.),
4109:The tail bound for the cases when
3555:
3351:
3262:
1471:{\displaystyle s^{2}\chi _{k}^{2}}
754:
372:
271:
215:
14:
16926:
15472:
15067:"The distribution of chi-squared"
14763:Journal of Educational Statistics
14239:Modified half-normal distribution
8923:has the chi-squared distribution
8777:has the chi-squared distribution
8245:{\displaystyle 1-{\frac {2}{9k}}}
1995:for stratified contingency tables
16883:
16874:
16873:
14823:Random Structures and Algorithms
14182:
14147:multivariate normal distribution
13551:cumulative distribution function
12210:of chi-square random variables.
10768:is chi-squared distributed with
9869:Inverse-chi-squared distribution
8379:
8280:{\displaystyle {\frac {2}{9k}}.}
7350:{\displaystyle \mathbb {R} ^{n}}
7321:{\displaystyle v\sim N(0,1)^{n}}
7109:
6840:degrees of freedom are given by
5832:chi-squared variables of degree
5755:is chi-squared distributed with
3483:cumulative distribution function
3439:Cumulative distribution function
1684:central chi-squared distribution
51:
49:Cumulative distribution function
39:
15302:
15279:
15246:
15226:
15211:
15158:
15119:
15058:
14973:
14947:Box, Hunter and Hunter (1978).
14940:
14891:
14871:
14855:
13102:
12753:{\displaystyle X_{1},...,X_{n}}
10967:is chi-square distributed with
10362:generalized normal distribution
8390:needs additional citations for
8017:removes much of the asymmetry.
7431:{\displaystyle n^{1/2+\alpha }}
7328:is a gaussian random vector in
6350:{\displaystyle {\overline {X}}}
6016:
5095:{\displaystyle b_{2},...,b_{n}}
4310:{\displaystyle Z_{1},...,Z_{n}}
3670:lower incomplete gamma function
3387:closed-form values for integer
1947:
1435:scaled chi-squared distribution
697:{\displaystyle {\frac {12}{k}}}
662:{\displaystyle {\sqrt {8/k}}\,}
181:(known as "degrees of freedom")
125:{\displaystyle \chi _{k}^{2}\!}
15197:10.1088/1751-8113/46/50/505202
14807:
14704:
14662:
14568:
14525:
14513:
14449:
14437:
14369:
14341:
14289:
14283:
14260:
14248:
13062:
13031:
12870:
12836:
12792:
12773:
12006:is not chi-square distributed.
11727:are statistically independent.
11521:
11495:
11070:
11058:
10755:
10743:
10721:
10708:
10453:
10431:
10289:
10268:
10219:
10207:
10147:
10141:
10104:
10092:
10052:
10008:
9698:
9692:
9607:
9601:
9520:
9508:
9469:
9449:
9365:
9351:
9191:
9156:
9012:
9000:
8901:
8859:
8840:
8746:
8705:
8679:
8632:with non-centrality parameter
8615:
8609:
8542:
8530:
8491:
8467:
8444:
8289:Wilson–Hilferty transformation
8162:
8156:
8119:{\displaystyle {\sqrt {2k-1}}}
8053:
8047:
7979:{\displaystyle \ln(\chi ^{2})}
7973:
7960:
7845:
7833:
7793:
7787:
7653:
7633:
7626:can be chosen as any value in
7564:
7561:
7473:
7451:
7309:
7296:
7264:
7255:
7243:
7215:
7188:
7179:
7167:
7130:
7086:
7074:
6932:
6911:
6905:
6893:
6890:
6878:
6869:
6856:
6791:
6785:
6773:
6759:
6750:
6747:
6741:
6732:
6700:
6694:
6637:
6631:
6486:
6473:
6461:
6448:
6247:
6204:
6193:
6051:
6045:
5804:
5399:
5392:
5370:
5329:
5312:
5303:
5290:
5259:
5250:
5200:
5162:
5149:, which can be chosen so that
4988:
4910:
4830:
4818:
4805:
4771:
4740:
4673:
4666:
4644:
4600:
4554:
4537:
4528:
4438:
4382:
4375:
4353:
4199:
4176:
4170:
4154:
4077:
4054:
4048:
4032:
3905:
3892:
3835:
3822:
3766:
3753:
3694:
3682:
3655:
3643:
3571:
3558:
3550:
3523:
3511:
3498:
3368:
3354:
3179:
3166:
2850:
2823:
2739:
2717:
2691:
2675:
2604:
2592:
2560:
2548:
2487:
2471:
2282:. Ramsey shows that the exact
2149:
2137:
1864:
1858:
1720:
1650:
1631:
1593:
1552:
1478:is a reparametrization of the
1417:
1405:
1327:
1296:
1042:
1020:
998:{\displaystyle (1-2it)^{-k/2}}
975:
956:
889:
873:
582:
564:
389:
375:
288:
274:
218:
203:
92:{\displaystyle \chi ^{2}(k)\;}
85:
79:
1:
15348:10.1080/03610926.2021.1934700
14740:. Boca Raton, FL: CRC Press.
14562:
14229:Reduced chi-squared statistic
14133:, for which he developed his
12693:problems via its role in the
12267:The chi-squared distribution
8948:{\displaystyle \chi _{1}^{2}}
7947:The sampling distribution of
7905:{\displaystyle {\sqrt {8/k}}}
5576:
4243:
2776:would generalize to the form
1931:that specifies the number of
1225:{\displaystyle \chi _{k}^{2}}
1200:The chi-squared distribution
589:{\displaystyle \max(k-2,0)\;}
14951:Statistics for experimenters
14801:The Chi-squared Distribution
12942:
12864:
10548:A chi-squared variable with
10533:Noncentral beta distribution
10364:(version 1) with parameters
8453:{\displaystyle k\to \infty }
8207:{\displaystyle {\sqrt{X/k}}}
8126:and unit variance (1922, by
8086:{\displaystyle {\sqrt {2X}}}
7280:One consequence is that, if
7022:
6342:
6180:
5907:
5688:
5620:
3661:{\displaystyle \gamma (s,t)}
3151:probability density function
3145:Probability density function
3024:{\displaystyle E_{i}=Np_{i}}
2155:{\displaystyle Z\sim N(0,1)}
1993:Cochran–Mantel–Haenszel test
37:Probability density function
7:
15437:Encyclopedia of Mathematics
14736:Westfall, Peter H. (2013).
14266:{\displaystyle (0,\infty )}
14234:Wilks's lambda distribution
14175:
13402:noncentral chi distribution
12665:Occurrence and applications
12138:, then the distribution of
11999:{\displaystyle X_{1}+X_{2}}
7682:) and relative difference (
4897:is the identity matrix and
4015:{\displaystyle 0<z<1}
3979:{\displaystyle z\equiv x/k}
2566:{\displaystyle N=Np+N(1-p)}
1360:is the shape parameter and
23:. For the music group, see
10:
16931:
16707:Wrapped asymmetric Laplace
15678:Extended negative binomial
15497:², for a normal population
15432:"Chi-squared distribution"
15144:10.1109/TASLP.2017.2757601
15072:Proc. Natl. Acad. Sci. USA
14224:Pearson's chi-squared test
14135:Pearson's chi-squared test
14109:
13615:
13493:magnetic resonance imaging
12240:
12222:
11966:are not independent, then
8651:{\displaystyle \lambda =0}
8296:normalizing transformation
7031:are readily obtained by a
6405:
4919:{\displaystyle {\bar {1}}}
4609:{\displaystyle {\bar {Z}}}
4252:
3709:regularized gamma function
18:
16869:
16803:
16761:
16662:
16498:
16476:
16467:
16366:Generalized extreme value
16351:
16186:
16146:Relativistic Breit–Wigner
15862:
15759:
15750:
15643:
15563:
15554:
15543:Probability distributions
14868:, retrieved Feb. 11, 2009
14026:
13635:{\displaystyle \chi ^{2}}
13600:{\displaystyle \chi ^{2}}
13004:The box below shows some
12305:is a special case of the
12214:Chi-squared distributions
11035:{\displaystyle p\times p}
10888:{\displaystyle k\times k}
10793:statistically independent
10542:Noncentral t-distribution
8006:{\displaystyle \chi ^{2}}
3374:{\textstyle \Gamma (k/2)}
2934:{\displaystyle \chi ^{2}}
2907:{\displaystyle \chi ^{2}}
2610:{\displaystyle N=m+(N-m)}
1668:probability distributions
1232:is a special case of the
1139:{\displaystyle \chi ^{2}}
1013:
1008:
949:
944:
866:
861:
712:
707:
677:
672:
636:
631:
604:
599:
554:
549:
480:
475:
452:
447:
359:
354:
240:
235:
190:
185:
140:
135:
62:
59:
47:
35:
14904:The Annals of Statistics
14862:Chi-squared distribution
14799:Lancaster, H.O. (1969),
14116:Friedrich Robert Helmert
13613:Degrees of freedom (df)
13571:statistical significance
13114:chi-squared distribution
12687:Student's t-distribution
12518:exponential distribution
12429:exponential distribution
12259:is an arbitrary matrix.
10524:Student's t-distribution
10518:Student's t-distribution
9380:exponential distribution
9130:{\displaystyle c>0\,}
7357:, then as the dimension
6643:{\displaystyle \psi (x)}
5571:which proves the claim.
3911:{\displaystyle F(x;\,k)}
3431:degrees of freedom, see
1959:exponential distribution
1112:chi-squared distribution
16361:Generalized chi-squared
16305:Normal-inverse Gaussian
15404:Elderton, William Palin
15167:J. Phys. A: Math. Theor
14555:Fox–Wright Psi function
12202:is a special case of a
11123:-vector independent of
11009:{\displaystyle \Sigma }
10960:{\displaystyle Y^{T}AY}
7753:{\displaystyle k>50}
7659:{\displaystyle (0,1/2)}
7619:{\displaystyle \alpha }
6083:{\displaystyle \alpha }
5885:{\displaystyle \theta }
5865:{\displaystyle \alpha }
4325:random variables, then
3447:Chernoff bound for the
2188:{\displaystyle Q=Z^{2}}
1678:and in construction of
1373:{\displaystyle \theta }
1353:{\displaystyle \alpha }
16673:Univariate (circular)
16234:Generalized hyperbolic
15663:Conway–Maxwell–Poisson
15653:Beta negative binomial
15464:10.1093/biomet/10.1.85
15424:10.1093/biomet/1.2.155
15316:See also Jeff Miller,
15094:10.1073/pnas.17.12.684
14917:10.1214/aos/1015957395
14547:
14421:
14267:
14149:with the Greek letter
13636:
13601:
13537:
13480:
13434:
13391:
13330:
13287:
13243:
13202:
13143:
13093:
12995:
12980:
12917:
12835:
12799:
12754:
12655:
12627:
12599:
12579:
12559:
12510:
12463:
12414:
12362:
12299:
12196:
12171:
12132:
12068:
12000:
11960:
11933:
11906:
11828:
11776:
11721:
11669:
11617:
11528:
11469:
11443:
11249:
11189:
11137:
11117:
11097:
11077:
11036:
11010:
10987:
10961:
10925:
10889:
10863:
10840:
10820:
10791:The sum of squares of
10782:
10762:
10689:
10669:
10649:
10629:
10609:
10582:
10562:
10503:
10423:
10390:
10354:
10325:
10260:
10227:
10173:
10112:
10060:
9966:
9920:
9861:
9796:
9756:
9706:
9665:
9615:
9574:
9527:
9476:
9424:
9372:
9326:
9281:
9244:
9199:
9131:
9104:
9047:
8949:
8917:
8867:
8822:As a special case, if
8810:
8771:
8712:
8652:
8622:
8550:
8454:
8355:
8281:
8246:
8208:
8169:
8120:
8087:
8060:
8007:
7980:
7938:
7906:
7867:
7820:
7800:
7754:
7728:
7708:
7687:
7660:
7620:
7600:
7432:
7391:
7371:
7351:
7322:
7271:
7195:
7100:
7013:
6834:
6798:
6713:
6675:
6644:
6612:
6396:
6351:
6324:
6300:
6280:
6254:
6160:
6116:
6084:
6058:
5945:
5886:
5866:
5846:
5823:
5795:
5749:
5697:
5656:
5629:
5565:
5369:
5336:
5207:
5143:
5123:
5096:
5044:
5018:
4998:
4940:
4920:
4891:
4890:{\displaystyle 1\!\!1}
4866:
4711:
4643:
4616:their average. Then
4610:
4581:
4561:
4495:
4477:
4419:
4352:
4311:
4223:
4129:
4128:{\displaystyle z>1}
4101:
4016:
3980:
3932:
3918:for other small, even
3912:
3876:
3803:
3734:
3701:
3662:
3624:
3478:
3471:
3425:
3401:
3375:
3335:
3135:
3112:
3092:
3065:
3045:
3025:
2982:
2962:
2935:
2908:
2876:
2819:
2763:
2643:
2611:
2567:
2514:
2436:
2404:
2384:
2364:
2341:
2267:likelihood ratio tests
2256:
2209:
2189:
2156:
2115:
2095:
2075:
1918:
1813:
1790:
1672:inferential statistics
1657:
1600:
1528:
1472:
1424:
1374:
1354:
1334:
1272:
1226:
1185:
1162:
1140:
1091:
999:
935:
852:
698:
663:
622:
590:
540:
466:
438:
345:
226:
175:
126:
93:
16718:Bivariate (spherical)
16216:Kaniadakis κ-Gaussian
14548:
14422:
14268:
14030:-value (probability)
13637:
13602:
13538:
13499:Computational methods
13481:
13414:
13392:
13310:
13288:
13223:
13203:
13123:
13094:
12996:
12960:
12918:
12815:
12800:
12755:
12685:line via its role in
12656:
12628:
12600:
12580:
12560:
12511:
12464:
12415:
12363:
12300:
12197:
12151:
12133:
12069:
12001:
11961:
11959:{\displaystyle X_{2}}
11934:
11932:{\displaystyle X_{1}}
11907:
11829:
11777:
11722:
11670:
11618:
11529:
11470:
11444:
11250:
11190:
11138:
11118:
11098:
11078:
11037:
11011:
10988:
10962:
10926:
10890:
10864:
10841:
10821:
10783:
10763:
10690:
10670:
10650:
10630:
10610:
10583:
10563:
10504:
10403:
10391:
10355:
10353:{\displaystyle X_{i}}
10326:
10240:
10228:
10174:
10113:
10061:
9972:are independent then
9967:
9921:
9862:
9797:
9757:
9707:
9666:
9623:Rayleigh distribution
9616:
9575:
9528:
9477:
9425:
9373:
9327:
9282:
9245:
9200:
9132:
9105:
9048:
8950:
8918:
8868:
8811:
8772:
8713:
8653:
8623:
8551:
8455:
8371:Related distributions
8356:
8287:This is known as the
8282:
8247:
8209:
8170:
8121:
8088:
8061:
8015:logarithmic transform
8008:
7981:
7939:
7907:
7868:
7821:
7801:
7755:
7729:
7709:
7692:central limit theorem
7677:
7670:Asymptotic properties
7661:
7621:
7601:
7433:
7392:
7372:
7352:
7323:
7272:
7196:
7101:
7014:
6835:
6799:
6714:
6676:
6661:for a random variate
6645:
6613:
6397:
6352:
6325:
6301:
6281:
6264:central limit theorem
6255:
6161:
6117:
6085:
6059:
5925:
5887:
5867:
5847:
5824:
5796:
5750:
5698:
5657:
5655:{\displaystyle k_{i}}
5630:
5566:
5349:
5337:
5208:
5144:
5124:
5122:{\displaystyle b_{1}}
5097:
5045:
5019:
4999:
4941:
4926:the all ones vector.
4921:
4892:
4867:
4691:
4623:
4611:
4582:
4562:
4496:
4457:
4420:
4332:
4312:
4224:
4130:
4102:
4017:
3981:
3933:
3913:
3877:
3804:
3735:
3714:In a special case of
3702:
3663:
3625:
3472:
3446:
3426:
3402:
3376:
3336:
3136:
3113:
3093:
3091:{\displaystyle p_{i}}
3071:in the population is
3066:
3046:
3026:
2983:
2963:
2961:{\displaystyle O_{i}}
2936:
2909:
2877:
2799:
2764:
2644:
2642:{\displaystyle q=1-p}
2612:
2568:
2515:
2437:
2435:{\displaystyle q=1-p}
2405:
2385:
2365:
2342:
2257:
2210:
2190:
2157:
2116:
2096:
2076:
2052:central limit theorem
2048:sampling distribution
1981:Likelihood-ratio test
1919:
1814:
1770:
1658:
1601:
1529:
1473:
1425:
1375:
1355:
1335:
1273:
1227:
1186:
1163:
1141:
1092:
1000:
936:
853:
699:
664:
623:
591:
541:
467:
439:
346:
227:
176:
127:
94:
16783:Dirac delta function
16730:Bivariate (toroidal)
16687:Univariate von Mises
16558:Multivariate Laplace
16450:Shifted log-logistic
15799:Continuous Bernoulli
15021:Annals of Statistics
14431:
14277:
14245:
13619:
13584:
13580:-values matching to
13527:
13409:
13305:
13220:
13120:
13012:
12927:
12812:
12767:
12712:
12691:analysis of variance
12637:
12633:and scale parameter
12609:
12589:
12569:
12531:
12473:
12435:
12372:
12313:
12271:
12142:
12078:
12032:
11970:
11943:
11916:
11838:
11786:
11734:
11679:
11627:
11538:
11483:
11459:
11262:
11199:
11147:
11127:
11107:
11087:
11046:
11020:
11000:
10993:degrees of freedom.
10971:
10938:
10909:
10873:
10853:
10830:
10810:
10788:degrees of freedom.
10772:
10699:
10679:
10659:
10648:{\displaystyle \mu }
10639:
10619:
10599:
10572:
10552:
10400:
10368:
10337:
10237:
10185:
10126:
10120:uniform distribution
10077:
9976:
9930:
9884:
9876:Pearson distribution
9806:
9768:
9720:
9714:Maxwell distribution
9677:
9629:
9586:
9537:
9493:
9434:
9393:
9336:
9298:
9254:
9216:
9141:
9114:
9075:
8964:
8927:
8877:
8826:
8781:
8722:
8665:
8636:
8566:
8464:
8438:
8399:improve this article
8302:
8256:
8218:
8179:
8137:
8097:
8070:
8028:
7990:
7951:
7937:{\displaystyle 12/k}
7920:
7884:
7830:
7810:
7768:
7738:
7718:
7698:
7630:
7610:
7442:
7401:
7381:
7361:
7332:
7284:
7206:
7121:
7042:
6847:
6824:
6723:
6685:
6665:
6625:
6421:
6412:differential entropy
6361:
6334:
6323:{\displaystyle 2\,k}
6310:
6290:
6270:
6172:
6126:
6094:
6074:
5899:
5876:
5856:
5836:
5813:
5801:degrees of freedom.
5759:
5707:
5666:
5639:
5585:
5346:
5217:
5153:
5133:
5106:
5054:
5028:
5008:
4950:
4946:has one eigenvector
4930:
4901:
4876:
4620:
4591:
4571:
4512:
4429:
4329:
4269:
4142:
4113:
4026:
3994:
3956:
3947:statistical packages
3922:
3886:
3816:
3747:
3718:
3676:
3637:
3492:
3470:{\displaystyle k=10}
3455:
3415:
3391:
3348:
3160:
3125:
3102:
3075:
3055:
3035:
2992:
2972:
2945:
2918:
2891:
2783:
2656:
2621:
2577:
2527:
2452:
2414:
2394:
2374:
2354:
2297:
2271:Neyman–Pearson lemma
2219:
2199:
2166:
2125:
2105:
2101:and the variance is
2085:
2065:
1989:in survival analysis
1833:
1758:
1680:confidence intervals
1610:
1538:
1490:
1486:. Specifically if
1484:Wishart distribution
1440:
1384:
1364:
1344:
1282:
1244:
1240:. Specifically if
1238:Wishart distribution
1204:
1175:
1152:
1123:
1017:
953:
870:
716:
681:
640:
621:{\displaystyle 2k\;}
608:
558:
484:
456:
363:
244:
194:
144:
103:
66:
16910:Normal distribution
16831:Natural exponential
16736:Bivariate von Mises
16702:Wrapped exponential
16568:Multivariate stable
16563:Multivariate normal
15884:Benktander 2nd kind
15879:Benktander 1st kind
15668:Discrete phase-type
15388:. New York: Wiley.
15299:, 1876, pp. 192–219
15189:2013JPhA...46X5202B
15085:1931PNAS...17..684W
13061:
12912:
12654:{\displaystyle 1/2}
12626:{\displaystyle k/2}
12554:
12525:Erlang distribution
12458:
12294:
11901:
11823:
11771:
11716:
11664:
11435:
10986:{\displaystyle k-n}
10924:{\displaystyle k-n}
10528:normal distribution
10513:Pareto distribution
10497:
10319:
10167:
9960:
9914:
9855:
9791:
9750:
9659:
9569:
9484:Erlang distribution
9419:
9321:
9239:
9098:
9042:
8944:
8805:
8608:
8583:
8558:normal distribution
8521:
8484:
7762:normal distribution
7606:where the exponent
6444:
6306:, and its variance
6286:the expectation is
6199:
5809:The sample mean of
5557:
5527:
5497:
5102:(all orthogonal to
5043:{\displaystyle n-1}
4726:
4414:
3733:{\displaystyle k=2}
3700:{\textstyle P(s,t)}
2280:Fisher's exact test
2251:
1968:of independence in
1955:normal distribution
1910:
1805:
1523:
1482:and the univariate
1467:
1267:
1236:and the univariate
1221:
1197:random variables.
120:
32:
16486:Rectified Gaussian
16371:Generalized Pareto
16229:Generalized normal
16101:Matrix-exponential
15238:2013-11-18 at the
15044:10.1214/15-aos1407
14608:Abramowitz, Milton
14543:
14530:
14417:
14263:
14209:Gamma distribution
14190:Mathematics portal
14129:in the context of
13632:
13597:
13533:
13476:
13387:
13283:
13198:
13089:
13047:
12991:
12913:
12892:
12795:
12750:
12677:and in estimating
12673:, for instance in
12651:
12623:
12595:
12575:
12555:
12540:
12506:
12459:
12444:
12410:
12358:
12307:gamma distribution
12295:
12280:
12192:
12128:
12064:
12024:Linear combination
11996:
11956:
11929:
11902:
11867:
11824:
11802:
11772:
11750:
11717:
11695:
11665:
11643:
11613:
11524:
11465:
11439:
11421:
11245:
11185:
11133:
11113:
11093:
11073:
11032:
11006:
10983:
10957:
10921:
10885:
10859:
10836:
10816:
10778:
10758:
10685:
10675:covariance matrix
10665:
10645:
10625:
10605:
10592:random variables.
10578:
10558:
10499:
10472:
10386:
10350:
10321:
10302:
10223:
10169:
10153:
10108:
10056:
10050:
10028:
9997:
9962:
9939:
9916:
9893:
9857:
9841:
9819:
9792:
9777:
9752:
9736:
9702:
9661:
9645:
9611:
9570:
9552:
9523:
9472:
9420:
9402:
9384:gamma distribution
9368:
9322:
9307:
9277:
9240:
9225:
9207:gamma distribution
9195:
9127:
9100:
9084:
9066:degrees of freedom
9043:
9028:
8945:
8930:
8913:
8908:
8863:
8806:
8784:
8767:
8753:
8708:
8648:
8618:
8587:
8569:
8546:
8470:
8450:
8351:
8277:
8242:
8204:
8165:
8116:
8083:
8056:
8003:
7976:
7934:
7902:
7863:
7816:
7796:
7750:
7724:
7704:
7688:
7656:
7616:
7596:
7428:
7387:
7367:
7347:
7318:
7267:
7191:
7096:
7009:
6830:
6816:Noncentral moments
6794:
6709:
6671:
6640:
6608:
6430:
6392:
6347:
6320:
6296:
6276:
6250:
6156:
6112:
6080:
6054:
5882:
5862:
5842:
5819:
5791:
5745:
5693:
5652:
5625:
5561:
5537:
5513:
5483:
5332:
5203:
5139:
5129:) with eigenvalue
5119:
5092:
5040:
5014:
4994:
4980:
4936:
4916:
4887:
4862:
4797:
4712:
4606:
4577:
4557:
4491:
4415:
4394:
4307:
4219:
4125:
4097:
4012:
3976:
3928:
3908:
3872:
3799:
3730:
3697:
3658:
3620:
3479:
3467:
3421:
3397:
3371:
3331:
3326:
3285:
3131:
3108:
3088:
3061:
3041:
3021:
2978:
2958:
2931:
2904:
2872:
2759:
2639:
2607:
2563:
2510:
2432:
2400:
2380:
2360:
2337:
2252:
2237:
2205:
2185:
2152:
2111:
2091:
2071:
1970:contingency tables
1933:degrees of freedom
1914:
1896:
1809:
1791:
1711:standard deviation
1676:hypothesis testing
1653:
1596:
1524:
1509:
1480:gamma distribution
1468:
1453:
1420:
1370:
1350:
1330:
1268:
1253:
1234:gamma distribution
1222:
1207:
1181:
1169:degrees of freedom
1158:
1136:
1104:probability theory
1087:
995:
931:
848:
846:
694:
659:
618:
586:
536:
462:
434:
341:
222:
171:
122:
106:
89:
30:
16897:
16896:
16494:
16493:
16463:
16462:
16354:whose type varies
16300:Normal (Gaussian)
16254:Hyperbolic secant
16203:Exponential power
16106:Maxwell–Boltzmann
15854:Wigner semicircle
15746:
15745:
15718:Parabolic fractal
15708:Negative binomial
15395:978-0-471-17912-2
14966:978-0-471-09315-2
14955:. Wiley. p.
14887:978-0-387-34657-1
14835:10.1002/rsa.10073
14747:978-1-4665-1210-8
14722:978-0-07-042864-5
14697:978-0-471-58495-7
14631:978-0-486-61272-0
14612:Stegun, Irene Ann
14502:
14415:
14406:
14405:
14391:
14170:covariance matrix
14141:), collected in (
14071:quantile function
14067:
14066:
13489:
13488:
13474:
13462:
13385:
13373:
13271:
13186:
12958:
12945:
12867:
12675:chi-squared tests
12598:{\displaystyle X}
12578:{\displaystyle k}
12500:
12397:
12351:
12338:
11611:
11468:{\displaystyle Y}
11416:
11402:
11369:
11331:
11298:
11136:{\displaystyle X}
11116:{\displaystyle p}
11096:{\displaystyle w}
10900:idempotent matrix
10862:{\displaystyle A}
10839:{\displaystyle k}
10819:{\displaystyle Y}
10781:{\displaystyle k}
10688:{\displaystyle C}
10668:{\displaystyle k}
10628:{\displaystyle k}
10608:{\displaystyle Y}
10581:{\displaystyle k}
10561:{\displaystyle k}
10467:
10297:
10068:beta distribution
10049:
10027:
9996:
9836:
9818:
9262:
8886:
8731:
8526:
8522:
8511:
8507:
8431:
8430:
8423:
8334:
8272:
8240:
8202:
8114:
8081:
7900:
7861:
7819:{\displaystyle k}
7727:{\displaystyle k}
7707:{\displaystyle k}
7390:{\displaystyle n}
7370:{\displaystyle n}
7241:
7156:
7004:
6997:
6973:
6833:{\displaystyle k}
6674:{\displaystyle X}
6599:
6575:
6542:
6507:
6390:
6345:
6299:{\displaystyle k}
6279:{\displaystyle k}
6200:
6183:
6020:
5923:
5910:
5845:{\displaystyle k}
5822:{\displaystyle n}
5691:
5623:
5536:
5530:
5482:
5476:
5442:
5436:
5416:
5410:
5395:
5315:
5262:
5142:{\displaystyle 1}
5017:{\displaystyle 0}
4991:
4978:
4977:
4939:{\displaystyle M}
4913:
4844:
4838:
4821:
4808:
4795:
4760:
4754:
4743:
4690:
4684:
4669:
4603:
4580:{\displaystyle n}
4540:
4455:
4441:
4378:
4255:Cochran's theorem
4249:Cochran's theorem
3931:{\displaystyle k}
3849:
3610:
3596:
3575:
3569:
3548:
3534:
3424:{\displaystyle k}
3400:{\displaystyle k}
3319:
3284:
3277:
3134:{\displaystyle n}
3111:{\displaystyle n}
3064:{\displaystyle i}
3044:{\displaystyle i}
2981:{\displaystyle i}
2870:
2757:
2709:
2508:
2403:{\displaystyle p}
2383:{\displaystyle N}
2363:{\displaystyle m}
2335:
2333:
2236:
2230:
2224:
2208:{\displaystyle Q}
2114:{\displaystyle 1}
2094:{\displaystyle 0}
2074:{\displaystyle Z}
2046:, increases, the
1983:for nested models
1895:
1889:
1883:
1880:
1876:
1872:
1869:
1847:
1841:
1766:
1695:chi-squared tests
1623:
1569:
1550:
1397:
1313:
1294:
1184:{\displaystyle k}
1161:{\displaystyle k}
1100:
1099:
1084:
1065:
928:
912:
838:
816:
772:
731:
692:
656:
519:
465:{\displaystyle k}
426:
412:
393:
292:
170:
167:
16922:
16887:
16886:
16877:
16876:
16816:Compound Poisson
16791:
16779:
16748:von Mises–Fisher
16744:
16732:
16720:
16682:Circular uniform
16678:
16598:
16542:
16513:
16474:
16473:
16376:Marchenko–Pastur
16239:Geometric stable
16156:Truncated normal
16049:Inverse Gaussian
15955:Hyperexponential
15794:Beta rectangular
15762:bounded interval
15757:
15756:
15625:Discrete uniform
15610:Poisson binomial
15561:
15560:
15536:
15529:
15522:
15513:
15512:
15467:
15445:
15427:
15399:
15368:
15367:
15342:(5): 1591–1613.
15327:
15321:
15308:R. L. Plackett,
15306:
15300:
15283:
15277:
15271:
15262:
15261:
15250:
15244:
15233:Chi-Squared Test
15230:
15224:
15215:
15209:
15208:
15182:
15162:
15156:
15155:
15123:
15117:
15116:
15106:
15096:
15062:
15056:
15055:
15037:
15028:(5): 2089–2097.
15015:
15006:
15005:
14977:
14971:
14970:
14954:
14944:
14938:
14937:
14919:
14895:
14889:
14875:
14869:
14859:
14853:
14852:
14850:
14849:
14820:
14811:
14805:
14804:
14796:
14787:
14786:
14758:
14752:
14751:
14733:
14727:
14726:
14708:
14702:
14701:
14683:
14672:
14666:
14660:
14659:
14614:, eds. (1983) .
14604:
14598:
14597:
14595:
14594:
14588:
14582:. Archived from
14581:
14572:
14552:
14550:
14549:
14544:
14542:
14538:
14531:
14508:
14504:
14503:
14495:
14473:
14472:
14463:
14462:
14457:
14426:
14424:
14423:
14418:
14416:
14414:
14413:
14412:
14408:
14407:
14401:
14397:
14392:
14384:
14372:
14359:
14358:
14334:
14333:
14318:
14317:
14313:
14296:
14272:
14270:
14269:
14264:
14241:with the pdf on
14199:Chi distribution
14192:
14187:
14186:
14167:
14156:
14106:from the table.
14098:
14091:
14087:
14083:
14076:
13641:
13639:
13638:
13633:
13631:
13630:
13610:
13609:
13606:
13604:
13603:
13598:
13596:
13595:
13542:
13540:
13539:
13534:
13517:
13509:
13485:
13483:
13482:
13477:
13475:
13473:
13472:
13467:
13463:
13461:
13460:
13451:
13450:
13441:
13433:
13428:
13413:
13396:
13394:
13393:
13388:
13386:
13384:
13383:
13378:
13374:
13372:
13371:
13362:
13361:
13360:
13348:
13347:
13337:
13329:
13324:
13309:
13298:chi distribution
13292:
13290:
13289:
13284:
13282:
13281:
13276:
13272:
13270:
13269:
13260:
13259:
13250:
13242:
13237:
13207:
13205:
13204:
13199:
13197:
13196:
13191:
13187:
13185:
13184:
13175:
13174:
13173:
13161:
13160:
13150:
13142:
13137:
13103:
13098:
13096:
13095:
13090:
13060:
13055:
13043:
13042:
13024:
13023:
13000:
12998:
12997:
12992:
12990:
12989:
12979:
12974:
12959:
12951:
12946:
12941:
12940:
12931:
12922:
12920:
12919:
12914:
12911:
12906:
12891:
12890:
12878:
12877:
12868:
12863:
12862:
12853:
12848:
12847:
12834:
12829:
12806:random variables
12804:
12802:
12801:
12796:
12791:
12790:
12759:
12757:
12756:
12751:
12749:
12748:
12724:
12723:
12699:random variables
12689:. It enters all
12660:
12658:
12657:
12652:
12647:
12632:
12630:
12629:
12624:
12619:
12604:
12602:
12601:
12596:
12584:
12582:
12581:
12576:
12564:
12562:
12561:
12556:
12553:
12548:
12515:
12513:
12512:
12507:
12505:
12501:
12493:
12468:
12466:
12465:
12460:
12457:
12452:
12423:
12419:
12417:
12416:
12411:
12409:
12405:
12398:
12390:
12367:
12365:
12364:
12359:
12357:
12353:
12352:
12344:
12339:
12331:
12304:
12302:
12301:
12296:
12293:
12288:
12258:
12254:
12250:
12201:
12199:
12198:
12193:
12191:
12190:
12181:
12180:
12170:
12165:
12137:
12135:
12134:
12129:
12127:
12126:
12118:
12109:
12108:
12090:
12089:
12073:
12071:
12070:
12065:
12063:
12062:
12044:
12043:
12018:
12005:
12003:
12002:
11997:
11995:
11994:
11982:
11981:
11965:
11963:
11962:
11957:
11955:
11954:
11938:
11936:
11935:
11930:
11928:
11927:
11911:
11909:
11908:
11903:
11900:
11895:
11894:
11893:
11881:
11880:
11863:
11862:
11850:
11849:
11833:
11831:
11830:
11825:
11822:
11817:
11816:
11815:
11798:
11797:
11781:
11779:
11778:
11773:
11770:
11765:
11764:
11763:
11746:
11745:
11726:
11724:
11723:
11718:
11715:
11710:
11709:
11708:
11691:
11690:
11674:
11672:
11671:
11666:
11663:
11658:
11657:
11656:
11639:
11638:
11622:
11620:
11619:
11614:
11612:
11610:
11609:
11608:
11607:
11597:
11592:
11591:
11590:
11579:
11578:
11577:
11576:
11566:
11561:
11560:
11559:
11548:
11533:
11531:
11530:
11525:
11520:
11519:
11507:
11506:
11474:
11472:
11471:
11466:
11448:
11446:
11445:
11440:
11434:
11429:
11417:
11415:
11414:
11413:
11408:
11404:
11403:
11401:
11400:
11391:
11390:
11381:
11370:
11368:
11367:
11358:
11357:
11348:
11337:
11333:
11332:
11330:
11329:
11320:
11319:
11310:
11299:
11297:
11296:
11287:
11286:
11277:
11266:
11254:
11252:
11251:
11246:
11211:
11210:
11194:
11192:
11191:
11186:
11178:
11177:
11159:
11158:
11142:
11140:
11139:
11134:
11122:
11120:
11119:
11114:
11102:
11100:
11099:
11094:
11082:
11080:
11079:
11074:
11041:
11039:
11038:
11033:
11015:
11013:
11012:
11007:
10992:
10990:
10989:
10984:
10966:
10964:
10963:
10958:
10950:
10949:
10930:
10928:
10927:
10922:
10894:
10892:
10891:
10886:
10868:
10866:
10865:
10860:
10845:
10843:
10842:
10837:
10825:
10823:
10822:
10817:
10787:
10785:
10784:
10779:
10767:
10765:
10764:
10759:
10742:
10741:
10729:
10728:
10694:
10692:
10691:
10686:
10674:
10672:
10671:
10666:
10654:
10652:
10651:
10646:
10634:
10632:
10631:
10626:
10614:
10612:
10611:
10606:
10587:
10585:
10584:
10579:
10567:
10565:
10564:
10559:
10508:
10506:
10505:
10500:
10496:
10491:
10487:
10468:
10463:
10462:
10461:
10456:
10444:
10443:
10434:
10425:
10422:
10417:
10395:
10393:
10392:
10387:
10359:
10357:
10356:
10351:
10349:
10348:
10330:
10328:
10327:
10322:
10318:
10313:
10298:
10293:
10292:
10281:
10280:
10271:
10262:
10259:
10254:
10232:
10230:
10229:
10224:
10197:
10196:
10178:
10176:
10175:
10170:
10166:
10161:
10117:
10115:
10114:
10109:
10065:
10063:
10062:
10057:
10051:
10045:
10044:
10035:
10029:
10023:
10022:
10013:
9998:
9995:
9981:
9971:
9969:
9968:
9963:
9959:
9954:
9953:
9952:
9925:
9923:
9922:
9917:
9913:
9908:
9907:
9906:
9866:
9864:
9863:
9858:
9854:
9849:
9837:
9834:
9820:
9811:
9801:
9799:
9798:
9793:
9790:
9785:
9761:
9759:
9758:
9753:
9749:
9744:
9732:
9731:
9711:
9709:
9708:
9703:
9670:
9668:
9667:
9662:
9658:
9653:
9641:
9640:
9620:
9618:
9617:
9612:
9579:
9577:
9576:
9571:
9568:
9563:
9532:
9530:
9529:
9524:
9481:
9479:
9478:
9473:
9465:
9429:
9427:
9426:
9421:
9418:
9413:
9377:
9375:
9374:
9369:
9361:
9331:
9329:
9328:
9323:
9320:
9315:
9289:chi distribution
9286:
9284:
9283:
9278:
9276:
9275:
9263:
9258:
9249:
9247:
9246:
9241:
9238:
9233:
9204:
9202:
9201:
9196:
9172:
9136:
9134:
9133:
9128:
9109:
9107:
9106:
9101:
9097:
9092:
9052:
9050:
9049:
9044:
9041:
9036:
9024:
9023:
8999:
8998:
8975:
8954:
8952:
8951:
8946:
8943:
8938:
8922:
8920:
8919:
8914:
8907:
8900:
8899:
8872:
8870:
8869:
8864:
8858:
8857:
8839:
8815:
8813:
8812:
8807:
8804:
8799:
8798:
8797:
8776:
8774:
8773:
8768:
8763:
8762:
8752:
8745:
8744:
8717:
8715:
8714:
8709:
8704:
8703:
8691:
8690:
8678:
8657:
8655:
8654:
8649:
8627:
8625:
8624:
8619:
8607:
8602:
8597:
8596:
8582:
8577:
8555:
8553:
8552:
8547:
8524:
8523:
8513:
8509:
8508:
8500:
8498:
8483:
8478:
8459:
8457:
8456:
8451:
8426:
8419:
8415:
8412:
8406:
8383:
8375:
8360:
8358:
8357:
8352:
8349:
8348:
8343:
8342:
8335:
8333:
8322:
8314:
8313:
8286:
8284:
8283:
8278:
8273:
8271:
8260:
8251:
8249:
8248:
8243:
8241:
8239:
8228:
8213:
8211:
8210:
8205:
8203:
8201:
8196:
8192:
8183:
8174:
8172:
8171:
8166:
8155:
8154:
8125:
8123:
8122:
8117:
8115:
8101:
8092:
8090:
8089:
8084:
8082:
8074:
8065:
8063:
8062:
8057:
8046:
8045:
8012:
8010:
8009:
8004:
8002:
8001:
7985:
7983:
7982:
7977:
7972:
7971:
7943:
7941:
7940:
7935:
7930:
7911:
7909:
7908:
7903:
7901:
7896:
7888:
7872:
7870:
7869:
7864:
7862:
7854:
7852:
7825:
7823:
7822:
7817:
7805:
7803:
7802:
7797:
7786:
7785:
7759:
7757:
7756:
7751:
7733:
7731:
7730:
7725:
7713:
7711:
7710:
7705:
7685:
7681:
7665:
7663:
7662:
7657:
7649:
7625:
7623:
7622:
7617:
7605:
7603:
7602:
7597:
7595:
7594:
7593:
7592:
7560:
7559:
7544:
7543:
7533:
7508:
7507:
7497:
7469:
7468:
7437:
7435:
7434:
7429:
7427:
7426:
7416:
7396:
7394:
7393:
7388:
7376:
7374:
7373:
7368:
7356:
7354:
7353:
7348:
7346:
7345:
7340:
7327:
7325:
7324:
7319:
7317:
7316:
7276:
7274:
7273:
7268:
7242:
7234:
7200:
7198:
7197:
7192:
7157:
7149:
7105:
7103:
7102:
7097:
7073:
7072:
7054:
7053:
7018:
7016:
7015:
7010:
7005:
7003:
7002:
6998:
6990:
6980:
6979:
6975:
6974:
6966:
6949:
6947:
6946:
6868:
6867:
6839:
6837:
6836:
6831:
6803:
6801:
6800:
6795:
6769:
6718:
6716:
6715:
6710:
6680:
6678:
6677:
6672:
6652:Digamma function
6649:
6647:
6646:
6641:
6617:
6615:
6614:
6609:
6604:
6600:
6592:
6581:
6577:
6576:
6568:
6552:
6548:
6547:
6543:
6535:
6508:
6500:
6443:
6438:
6401:
6399:
6398:
6393:
6391:
6386:
6378:
6373:
6372:
6356:
6354:
6353:
6348:
6346:
6338:
6329:
6327:
6326:
6321:
6305:
6303:
6302:
6297:
6285:
6283:
6282:
6277:
6259:
6257:
6256:
6251:
6243:
6228:
6227:
6185:
6184:
6176:
6165:
6163:
6162:
6157:
6155:
6154:
6138:
6137:
6121:
6119:
6118:
6113:
6089:
6087:
6086:
6081:
6063:
6061:
6060:
6055:
6044:
6043:
6031:
6030:
6021:
6018:
6015:
6011:
6007:
5987:
5955:
5954:
5944:
5939:
5924:
5916:
5911:
5903:
5891:
5889:
5888:
5883:
5871:
5869:
5868:
5863:
5851:
5849:
5848:
5843:
5828:
5826:
5825:
5820:
5800:
5798:
5797:
5792:
5790:
5789:
5771:
5770:
5754:
5752:
5751:
5746:
5744:
5743:
5725:
5724:
5702:
5700:
5699:
5694:
5692:
5687:
5676:
5661:
5659:
5658:
5653:
5651:
5650:
5634:
5632:
5631:
5626:
5624:
5619:
5608:
5597:
5596:
5570:
5568:
5567:
5562:
5556:
5551:
5534:
5528:
5526:
5521:
5496:
5491:
5480:
5474:
5463:
5462:
5452:
5451:
5440:
5434:
5426:
5425:
5414:
5408:
5407:
5406:
5397:
5396:
5388:
5382:
5381:
5368:
5363:
5341:
5339:
5338:
5333:
5317:
5316:
5308:
5302:
5301:
5277:
5276:
5264:
5263:
5255:
5249:
5248:
5235:
5234:
5212:
5210:
5209:
5204:
5199:
5198:
5174:
5173:
5148:
5146:
5145:
5140:
5128:
5126:
5125:
5120:
5118:
5117:
5101:
5099:
5098:
5093:
5091:
5090:
5066:
5065:
5049:
5047:
5046:
5041:
5023:
5021:
5020:
5015:
5004:with eigenvalue
5003:
5001:
5000:
4995:
4993:
4992:
4984:
4981:
4979:
4973:
4969:
4962:
4961:
4945:
4943:
4942:
4937:
4925:
4923:
4922:
4917:
4915:
4914:
4906:
4896:
4894:
4893:
4888:
4871:
4869:
4868:
4863:
4854:
4853:
4842:
4836:
4829:
4828:
4823:
4822:
4814:
4810:
4809:
4801:
4798:
4796:
4788:
4770:
4769:
4758:
4752:
4751:
4750:
4745:
4744:
4736:
4725:
4720:
4710:
4705:
4688:
4682:
4681:
4680:
4671:
4670:
4662:
4656:
4655:
4642:
4637:
4615:
4613:
4612:
4607:
4605:
4604:
4596:
4586:
4584:
4583:
4578:
4566:
4564:
4563:
4558:
4542:
4541:
4533:
4527:
4526:
4500:
4498:
4497:
4492:
4487:
4486:
4476:
4471:
4456:
4448:
4443:
4442:
4434:
4424:
4422:
4421:
4416:
4413:
4408:
4390:
4389:
4380:
4379:
4371:
4365:
4364:
4351:
4346:
4316:
4314:
4313:
4308:
4306:
4305:
4281:
4280:
4228:
4226:
4225:
4220:
4215:
4214:
4210:
4197:
4196:
4135:, similarly, is
4134:
4132:
4131:
4126:
4106:
4104:
4103:
4098:
4093:
4092:
4088:
4075:
4074:
4021:
4019:
4018:
4013:
3985:
3983:
3982:
3977:
3972:
3937:
3935:
3934:
3929:
3917:
3915:
3914:
3909:
3881:
3879:
3878:
3873:
3871:
3870:
3866:
3850:
3842:
3808:
3806:
3805:
3800:
3798:
3797:
3793:
3739:
3737:
3736:
3731:
3706:
3704:
3703:
3698:
3667:
3665:
3664:
3659:
3629:
3627:
3626:
3621:
3616:
3612:
3611:
3603:
3597:
3589:
3576:
3574:
3570:
3562:
3553:
3549:
3541:
3535:
3527:
3518:
3476:
3474:
3473:
3468:
3430:
3428:
3427:
3422:
3406:
3404:
3403:
3398:
3380:
3378:
3377:
3372:
3364:
3340:
3338:
3337:
3332:
3330:
3329:
3320:
3317:
3286:
3283:
3282:
3278:
3270:
3261:
3260:
3256:
3242:
3241:
3240:
3236:
3220:
3219:
3209:
3195:
3140:
3138:
3137:
3132:
3117:
3115:
3114:
3109:
3097:
3095:
3094:
3089:
3087:
3086:
3070:
3068:
3067:
3062:
3050:
3048:
3047:
3042:
3030:
3028:
3027:
3022:
3020:
3019:
3004:
3003:
2987:
2985:
2984:
2979:
2967:
2965:
2964:
2959:
2957:
2956:
2940:
2938:
2937:
2932:
2930:
2929:
2913:
2911:
2910:
2905:
2903:
2902:
2881:
2879:
2878:
2873:
2871:
2869:
2868:
2859:
2858:
2857:
2848:
2847:
2835:
2834:
2821:
2818:
2813:
2795:
2794:
2768:
2766:
2765:
2760:
2758:
2756:
2748:
2747:
2746:
2715:
2710:
2708:
2700:
2699:
2698:
2673:
2668:
2667:
2648:
2646:
2645:
2640:
2616:
2614:
2613:
2608:
2572:
2570:
2569:
2564:
2519:
2517:
2516:
2511:
2509:
2507:
2496:
2495:
2494:
2469:
2464:
2463:
2441:
2439:
2438:
2433:
2409:
2407:
2406:
2401:
2389:
2387:
2386:
2381:
2369:
2367:
2366:
2361:
2346:
2344:
2343:
2338:
2336:
2334:
2323:
2321:
2307:
2261:
2259:
2258:
2253:
2250:
2245:
2234:
2228:
2222:
2214:
2212:
2211:
2206:
2194:
2192:
2191:
2186:
2184:
2183:
2161:
2159:
2158:
2153:
2120:
2118:
2117:
2112:
2100:
2098:
2097:
2092:
2080:
2078:
2077:
2072:
2057:
2045:
1975:Chi-squared test
1966:Chi-squared test
1930:
1923:
1921:
1920:
1915:
1909:
1904:
1893:
1887:
1881:
1878:
1877:
1874:
1870:
1867:
1857:
1856:
1845:
1839:
1825:
1818:
1816:
1815:
1810:
1804:
1799:
1789:
1784:
1764:
1742:
1707:qualitative data
1662:
1660:
1659:
1654:
1643:
1642:
1630:
1629:
1624:
1621:
1605:
1603:
1602:
1597:
1592:
1591:
1570:
1562:
1551:
1548:
1533:
1531:
1530:
1525:
1522:
1517:
1508:
1507:
1477:
1475:
1474:
1469:
1466:
1461:
1452:
1451:
1429:
1427:
1426:
1421:
1404:
1403:
1398:
1395:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1314:
1306:
1295:
1292:
1277:
1275:
1274:
1269:
1266:
1261:
1231:
1229:
1228:
1223:
1220:
1215:
1190:
1188:
1187:
1182:
1167:
1165:
1164:
1159:
1145:
1143:
1142:
1137:
1135:
1134:
1096:
1094:
1093:
1088:
1085:
1080:
1066:
1063:
1061:
1060:
1056:
1004:
1002:
1001:
996:
994:
993:
989:
940:
938:
937:
932:
929:
921:
913:
910:
908:
907:
903:
857:
855:
854:
849:
847:
843:
839:
831:
822:
818:
817:
809:
789:
785:
781:
780:
779:
773:
765:
763:
762:
732:
724:
703:
701:
700:
695:
693:
685:
668:
666:
665:
660:
657:
652:
644:
627:
625:
624:
619:
595:
593:
592:
587:
545:
543:
542:
537:
534:
533:
528:
527:
520:
518:
507:
499:
498:
471:
469:
468:
463:
443:
441:
440:
435:
432:
428:
427:
419:
413:
405:
394:
392:
385:
367:
350:
348:
347:
342:
339:
338:
334:
318:
317:
307:
293:
291:
284:
270:
269:
265:
248:
231:
229:
228:
223:
180:
178:
177:
172:
168:
165:
164:
163:
158:
131:
129:
128:
123:
119:
114:
98:
96:
95:
90:
78:
77:
55:
43:
33:
29:
21:chi-squared test
16930:
16929:
16925:
16924:
16923:
16921:
16920:
16919:
16900:
16899:
16898:
16893:
16865:
16841:Maximum entropy
16799:
16787:
16775:
16765:
16757:
16740:
16728:
16716:
16671:
16658:
16595:Matrix-valued:
16592:
16538:
16509:
16501:
16490:
16478:
16469:
16459:
16353:
16347:
16264:
16190:
16188:
16182:
16111:Maxwell–Jüttner
15960:Hypoexponential
15866:
15864:
15863:supported on a
15858:
15819:Noncentral beta
15779:Balding–Nichols
15761:
15760:supported on a
15752:
15742:
15645:
15639:
15635:Zipf–Mandelbrot
15565:
15556:
15550:
15540:
15475:
15470:
15430:
15396:
15376:
15374:Further reading
15371:
15328:
15324:
15307:
15303:
15284:
15280:
15272:
15265:
15258:www.r-tutor.com
15252:
15251:
15247:
15240:Wayback Machine
15231:
15227:
15216:
15212:
15163:
15159:
15124:
15120:
15079:(12): 684–688.
15063:
15059:
15016:
15009:
14994:10.2307/2983618
14978:
14974:
14967:
14945:
14941:
14896:
14892:
14876:
14872:
14860:
14856:
14847:
14845:
14818:
14812:
14808:
14797:
14790:
14775:10.2307/1164752
14759:
14755:
14748:
14734:
14730:
14723:
14709:
14705:
14698:
14684:
14675:
14667:
14663:
14632:
14605:
14601:
14592:
14590:
14586:
14579:
14573:
14569:
14565:
14560:
14529:
14528:
14510:
14509:
14494:
14487:
14483:
14479:
14478:
14474:
14468:
14464:
14458:
14456:
14455:
14432:
14429:
14428:
14396:
14383:
14382:
14378:
14377:
14373:
14354:
14350:
14323:
14319:
14309:
14305:
14301:
14297:
14295:
14278:
14275:
14274:
14246:
14243:
14242:
14188:
14181:
14178:
14158:
14154:
14131:goodness of fit
14112:
14093:
14089:
14085:
14078:
14074:
13626:
13622:
13620:
13617:
13616:
13591:
13587:
13585:
13582:
13581:
13528:
13525:
13524:
13520:
13511:
13505:
13501:
13468:
13456:
13452:
13446:
13442:
13440:
13436:
13435:
13429:
13418:
13412:
13410:
13407:
13406:
13379:
13367:
13363:
13356:
13352:
13343:
13339:
13338:
13336:
13332:
13331:
13325:
13314:
13308:
13306:
13303:
13302:
13277:
13265:
13261:
13255:
13251:
13249:
13245:
13244:
13238:
13227:
13221:
13218:
13217:
13192:
13180:
13176:
13169:
13165:
13156:
13152:
13151:
13149:
13145:
13144:
13138:
13127:
13121:
13118:
13117:
13056:
13051:
13038:
13034:
13019:
13015:
13013:
13010:
13009:
12985:
12981:
12975:
12964:
12950:
12936:
12932:
12930:
12928:
12925:
12924:
12907:
12896:
12886:
12882:
12873:
12869:
12858:
12854:
12852:
12843:
12839:
12830:
12819:
12813:
12810:
12809:
12786:
12782:
12768:
12765:
12764:
12744:
12740:
12719:
12715:
12713:
12710:
12709:
12667:
12643:
12638:
12635:
12634:
12615:
12610:
12607:
12606:
12590:
12587:
12586:
12570:
12567:
12566:
12549:
12544:
12532:
12529:
12528:
12492:
12488:
12474:
12471:
12470:
12453:
12448:
12436:
12433:
12432:
12424:is an integer.
12421:
12389:
12388:
12384:
12373:
12370:
12369:
12343:
12330:
12329:
12325:
12314:
12311:
12310:
12289:
12284:
12272:
12269:
12268:
12265:
12256:
12252:
12248:
12245:
12239:
12227:
12221:
12216:
12186:
12182:
12176:
12172:
12166:
12155:
12143:
12140:
12139:
12119:
12114:
12113:
12104:
12100:
12085:
12081:
12079:
12076:
12075:
12058:
12054:
12039:
12035:
12033:
12030:
12029:
12026:
12016:
12013:
12011:Generalizations
11990:
11986:
11977:
11973:
11971:
11968:
11967:
11950:
11946:
11944:
11941:
11940:
11923:
11919:
11917:
11914:
11913:
11896:
11889:
11885:
11876:
11872:
11871:
11858:
11854:
11845:
11841:
11839:
11836:
11835:
11818:
11811:
11807:
11806:
11793:
11789:
11787:
11784:
11783:
11766:
11759:
11755:
11754:
11741:
11737:
11735:
11732:
11731:
11711:
11704:
11700:
11699:
11686:
11682:
11680:
11677:
11676:
11659:
11652:
11648:
11647:
11634:
11630:
11628:
11625:
11624:
11603:
11599:
11598:
11593:
11586:
11582:
11581:
11580:
11572:
11568:
11567:
11562:
11555:
11551:
11550:
11549:
11547:
11539:
11536:
11535:
11515:
11511:
11502:
11498:
11484:
11481:
11480:
11460:
11457:
11456:
11430:
11425:
11409:
11396:
11392:
11386:
11382:
11380:
11363:
11359:
11353:
11349:
11347:
11346:
11342:
11341:
11325:
11321:
11315:
11311:
11309:
11292:
11288:
11282:
11278:
11276:
11275:
11271:
11270:
11265:
11263:
11260:
11259:
11206:
11202:
11200:
11197:
11196:
11173:
11169:
11154:
11150:
11148:
11145:
11144:
11128:
11125:
11124:
11108:
11105:
11104:
11088:
11085:
11084:
11047:
11044:
11043:
11021:
11018:
11017:
11001:
10998:
10997:
10972:
10969:
10968:
10945:
10941:
10939:
10936:
10935:
10910:
10907:
10906:
10874:
10871:
10870:
10854:
10851:
10850:
10831:
10828:
10827:
10826:is a vector of
10811:
10808:
10807:
10773:
10770:
10769:
10734:
10730:
10724:
10720:
10700:
10697:
10696:
10680:
10677:
10676:
10660:
10657:
10656:
10640:
10637:
10636:
10620:
10617:
10616:
10600:
10597:
10596:
10590:standard normal
10573:
10570:
10569:
10553:
10550:
10549:
10492:
10483:
10476:
10457:
10452:
10451:
10439:
10435:
10430:
10426:
10424:
10418:
10407:
10401:
10398:
10397:
10369:
10366:
10365:
10344:
10340:
10338:
10335:
10334:
10314:
10306:
10288:
10276:
10272:
10267:
10263:
10261:
10255:
10244:
10238:
10235:
10234:
10192:
10188:
10186:
10183:
10182:
10162:
10157:
10127:
10124:
10123:
10078:
10075:
10074:
10040:
10036:
10033:
10018:
10014:
10011:
9985:
9979:
9977:
9974:
9973:
9955:
9948:
9944:
9943:
9931:
9928:
9927:
9909:
9902:
9898:
9897:
9885:
9882:
9881:
9850:
9845:
9824:
9809:
9807:
9804:
9803:
9786:
9781:
9769:
9766:
9765:
9745:
9740:
9727:
9723:
9721:
9718:
9717:
9678:
9675:
9674:
9654:
9649:
9636:
9632:
9630:
9627:
9626:
9587:
9584:
9583:
9564:
9556:
9538:
9535:
9534:
9494:
9491:
9490:
9461:
9435:
9432:
9431:
9414:
9406:
9394:
9391:
9390:
9357:
9337:
9334:
9333:
9316:
9311:
9299:
9296:
9295:
9271:
9267:
9257:
9255:
9252:
9251:
9234:
9229:
9217:
9214:
9213:
9168:
9142:
9139:
9138:
9115:
9112:
9111:
9093:
9088:
9076:
9073:
9072:
9037:
9032:
9019:
9015:
8976:
8971:
8970:
8965:
8962:
8961:
8939:
8934:
8928:
8925:
8924:
8895:
8891:
8890:
8878:
8875:
8874:
8853:
8849:
8835:
8827:
8824:
8823:
8800:
8793:
8789:
8788:
8782:
8779:
8778:
8758:
8754:
8740:
8736:
8735:
8723:
8720:
8719:
8699:
8695:
8686:
8682:
8674:
8666:
8663:
8662:
8637:
8634:
8633:
8603:
8598:
8589:
8588:
8578:
8573:
8567:
8564:
8563:
8512:
8499:
8494:
8479:
8474:
8465:
8462:
8461:
8439:
8436:
8435:
8427:
8416:
8410:
8407:
8396:
8384:
8373:
8367:
8344:
8338:
8337:
8336:
8326:
8321:
8309:
8308:
8303:
8300:
8299:
8264:
8259:
8257:
8254:
8253:
8232:
8227:
8219:
8216:
8215:
8197:
8188:
8184:
8182:
8180:
8177:
8176:
8150:
8146:
8138:
8135:
8134:
8100:
8098:
8095:
8094:
8073:
8071:
8068:
8067:
8041:
8037:
8029:
8026:
8025:
7997:
7993:
7991:
7988:
7987:
7967:
7963:
7952:
7949:
7948:
7926:
7921:
7918:
7917:
7914:excess kurtosis
7892:
7887:
7885:
7882:
7881:
7853:
7848:
7831:
7828:
7827:
7811:
7808:
7807:
7781:
7777:
7769:
7766:
7765:
7739:
7736:
7735:
7719:
7716:
7715:
7699:
7696:
7695:
7683:
7679:
7672:
7645:
7631:
7628:
7627:
7611:
7608:
7607:
7588:
7584:
7580:
7576:
7555:
7551:
7529:
7525:
7521:
7493:
7489:
7485:
7464:
7460:
7443:
7440:
7439:
7412:
7408:
7404:
7402:
7399:
7398:
7382:
7379:
7378:
7362:
7359:
7358:
7341:
7336:
7335:
7333:
7330:
7329:
7312:
7308:
7285:
7282:
7281:
7233:
7207:
7204:
7203:
7148:
7122:
7119:
7118:
7112:
7062:
7058:
7049:
7045:
7043:
7040:
7039:
7025:
6989:
6985:
6981:
6965:
6958:
6954:
6950:
6948:
6942:
6938:
6863:
6859:
6848:
6845:
6844:
6825:
6822:
6821:
6818:
6765:
6724:
6721:
6720:
6686:
6683:
6682:
6666:
6663:
6662:
6626:
6623:
6622:
6591:
6587:
6567:
6560:
6556:
6534:
6530:
6522:
6518:
6499:
6439:
6434:
6422:
6419:
6418:
6408:
6379:
6377:
6368:
6364:
6362:
6359:
6358:
6337:
6335:
6332:
6331:
6311:
6308:
6307:
6291:
6288:
6287:
6271:
6268:
6267:
6239:
6223:
6219:
6175:
6173:
6170:
6169:
6150:
6146:
6133:
6129:
6127:
6124:
6123:
6095:
6092:
6091:
6075:
6072:
6071:
6039:
6035:
6026:
6022:
6017:
6003:
5983:
5969:
5965:
5950:
5946:
5940:
5929:
5915:
5902:
5900:
5897:
5896:
5877:
5874:
5873:
5857:
5854:
5853:
5837:
5834:
5833:
5814:
5811:
5810:
5807:
5785:
5781:
5766:
5762:
5760:
5757:
5756:
5739:
5735:
5720:
5716:
5708:
5705:
5704:
5677:
5675:
5667:
5664:
5663:
5646:
5642:
5640:
5637:
5636:
5609:
5607:
5592:
5588:
5586:
5583:
5582:
5579:
5574:
5573:
5552:
5541:
5522:
5517:
5492:
5487:
5458:
5454:
5447:
5443:
5421:
5417:
5402:
5398:
5387:
5386:
5377:
5373:
5364:
5353:
5347:
5344:
5343:
5307:
5306:
5297:
5296:
5272:
5268:
5254:
5253:
5244:
5243:
5230:
5226:
5218:
5215:
5214:
5194:
5190:
5169:
5165:
5154:
5151:
5150:
5134:
5131:
5130:
5113:
5109:
5107:
5104:
5103:
5086:
5082:
5061:
5057:
5055:
5052:
5051:
5029:
5026:
5025:
5009:
5006:
5005:
4983:
4982:
4968:
4966:
4957:
4953:
4951:
4948:
4947:
4931:
4928:
4927:
4905:
4904:
4902:
4899:
4898:
4877:
4874:
4873:
4849:
4845:
4824:
4813:
4812:
4811:
4800:
4799:
4787:
4785:
4765:
4761:
4746:
4735:
4734:
4733:
4721:
4716:
4706:
4695:
4676:
4672:
4661:
4660:
4651:
4647:
4638:
4627:
4621:
4618:
4617:
4595:
4594:
4592:
4589:
4588:
4572:
4569:
4568:
4567:be a vector of
4532:
4531:
4522:
4521:
4513:
4510:
4509:
4482:
4478:
4472:
4461:
4447:
4433:
4432:
4430:
4427:
4426:
4409:
4398:
4385:
4381:
4370:
4369:
4360:
4356:
4347:
4336:
4330:
4327:
4326:
4323:standard normal
4301:
4297:
4276:
4272:
4270:
4267:
4266:
4257:
4251:
4246:
4206:
4202:
4198:
4186:
4182:
4143:
4140:
4139:
4114:
4111:
4110:
4084:
4080:
4076:
4064:
4060:
4027:
4024:
4023:
3995:
3992:
3991:
3988:Chernoff bounds
3968:
3957:
3954:
3953:
3923:
3920:
3919:
3887:
3884:
3883:
3862:
3855:
3851:
3841:
3817:
3814:
3813:
3789:
3782:
3778:
3748:
3745:
3744:
3719:
3716:
3715:
3677:
3674:
3673:
3638:
3635:
3634:
3602:
3588:
3587:
3583:
3561:
3554:
3540:
3526:
3519:
3517:
3493:
3490:
3489:
3456:
3453:
3452:
3441:
3416:
3413:
3412:
3392:
3389:
3388:
3360:
3349:
3346:
3345:
3325:
3324:
3316:
3314:
3305:
3304:
3290:
3269:
3265:
3252:
3248:
3244:
3243:
3232:
3225:
3221:
3205:
3201:
3197:
3196:
3193:
3186:
3185:
3161:
3158:
3157:
3147:
3126:
3123:
3122:
3103:
3100:
3099:
3082:
3078:
3076:
3073:
3072:
3056:
3053:
3052:
3036:
3033:
3032:
3015:
3011:
2999:
2995:
2993:
2990:
2989:
2973:
2970:
2969:
2952:
2948:
2946:
2943:
2942:
2925:
2921:
2919:
2916:
2915:
2898:
2894:
2892:
2889:
2888:
2864:
2860:
2853:
2849:
2843:
2839:
2830:
2826:
2822:
2820:
2814:
2803:
2790:
2786:
2784:
2781:
2780:
2749:
2742:
2738:
2716:
2714:
2701:
2694:
2690:
2674:
2672:
2663:
2659:
2657:
2654:
2653:
2622:
2619:
2618:
2578:
2575:
2574:
2528:
2525:
2524:
2497:
2490:
2486:
2470:
2468:
2459:
2455:
2453:
2450:
2449:
2415:
2412:
2411:
2395:
2392:
2391:
2375:
2372:
2371:
2355:
2352:
2351:
2322:
2308:
2306:
2298:
2295:
2294:
2246:
2241:
2220:
2217:
2216:
2200:
2197:
2196:
2179:
2175:
2167:
2164:
2163:
2126:
2123:
2122:
2106:
2103:
2102:
2086:
2083:
2082:
2066:
2063:
2062:
2055:
2043:
1950:
1943:
1928:
1905:
1900:
1873:
1852:
1848:
1834:
1831:
1830:
1823:
1800:
1795:
1785:
1774:
1759:
1756:
1755:
1749:standard normal
1741:
1732:
1726:
1723:
1699:goodness of fit
1638:
1634:
1625:
1620:
1619:
1611:
1608:
1607:
1587:
1583:
1561:
1547:
1539:
1536:
1535:
1518:
1513:
1503:
1499:
1491:
1488:
1487:
1462:
1457:
1447:
1443:
1441:
1438:
1437:
1399:
1394:
1393:
1385:
1382:
1381:
1365:
1362:
1361:
1345:
1342:
1341:
1305:
1291:
1283:
1280:
1279:
1262:
1257:
1245:
1242:
1241:
1216:
1211:
1205:
1202:
1201:
1195:standard normal
1176:
1173:
1172:
1153:
1150:
1149:
1130:
1126:
1124:
1121:
1120:
1079:
1064: for
1062:
1052:
1045:
1041:
1018:
1015:
1014:
985:
978:
974:
954:
951:
950:
920:
911: for
909:
899:
892:
888:
871:
868:
867:
845:
844:
830:
826:
808:
801:
797:
787:
786:
775:
774:
764:
758:
757:
750:
746:
733:
723:
719:
717:
714:
713:
684:
682:
679:
678:
674:Excess kurtosis
648:
643:
641:
638:
637:
609:
606:
605:
559:
556:
555:
529:
523:
522:
521:
511:
506:
494:
493:
485:
482:
481:
457:
454:
453:
418:
404:
403:
399:
381:
371:
366:
364:
361:
360:
330:
323:
319:
303:
299:
295:
280:
261:
257:
253:
252:
247:
245:
242:
241:
195:
192:
191:
159:
154:
153:
145:
142:
141:
115:
110:
104:
101:
100:
73:
69:
67:
64:
63:
50:
38:
28:
17:
12:
11:
5:
16928:
16918:
16917:
16912:
16895:
16894:
16892:
16891:
16881:
16870:
16867:
16866:
16864:
16863:
16858:
16853:
16848:
16843:
16838:
16836:Location–scale
16833:
16828:
16823:
16818:
16813:
16807:
16805:
16801:
16800:
16798:
16797:
16792:
16785:
16780:
16772:
16770:
16759:
16758:
16756:
16755:
16750:
16745:
16738:
16733:
16726:
16721:
16714:
16709:
16704:
16699:
16697:Wrapped Cauchy
16694:
16692:Wrapped normal
16689:
16684:
16679:
16668:
16666:
16660:
16659:
16657:
16656:
16655:
16654:
16649:
16647:Normal-inverse
16644:
16639:
16629:
16628:
16627:
16617:
16609:
16604:
16599:
16590:
16589:
16588:
16578:
16570:
16565:
16560:
16555:
16554:
16553:
16543:
16536:
16535:
16534:
16529:
16519:
16514:
16506:
16504:
16496:
16495:
16492:
16491:
16489:
16488:
16482:
16480:
16471:
16465:
16464:
16461:
16460:
16458:
16457:
16452:
16447:
16439:
16431:
16423:
16414:
16405:
16396:
16387:
16378:
16373:
16368:
16363:
16357:
16355:
16349:
16348:
16346:
16345:
16340:
16338:Variance-gamma
16335:
16330:
16322:
16317:
16312:
16307:
16302:
16297:
16289:
16284:
16283:
16282:
16272:
16267:
16262:
16256:
16251:
16246:
16241:
16236:
16231:
16226:
16218:
16213:
16205:
16200:
16194:
16192:
16184:
16183:
16181:
16180:
16178:Wilks's lambda
16175:
16174:
16173:
16163:
16158:
16153:
16148:
16143:
16138:
16133:
16128:
16123:
16118:
16116:Mittag-Leffler
16113:
16108:
16103:
16098:
16093:
16088:
16083:
16078:
16073:
16068:
16063:
16058:
16057:
16056:
16046:
16037:
16032:
16027:
16026:
16025:
16015:
16013:gamma/Gompertz
16010:
16009:
16008:
16003:
15993:
15988:
15983:
15982:
15981:
15969:
15968:
15967:
15962:
15957:
15947:
15946:
15945:
15935:
15930:
15925:
15924:
15923:
15922:
15921:
15911:
15901:
15896:
15891:
15886:
15881:
15876:
15870:
15868:
15865:semi-infinite
15860:
15859:
15857:
15856:
15851:
15846:
15841:
15836:
15831:
15826:
15821:
15816:
15811:
15806:
15801:
15796:
15791:
15786:
15781:
15776:
15771:
15765:
15763:
15754:
15748:
15747:
15744:
15743:
15741:
15740:
15735:
15730:
15725:
15720:
15715:
15710:
15705:
15700:
15695:
15690:
15685:
15680:
15675:
15670:
15665:
15660:
15655:
15649:
15647:
15644:with infinite
15641:
15640:
15638:
15637:
15632:
15627:
15622:
15617:
15612:
15607:
15606:
15605:
15598:Hypergeometric
15595:
15590:
15585:
15580:
15575:
15569:
15567:
15558:
15552:
15551:
15539:
15538:
15531:
15524:
15516:
15510:
15509:
15504:
15499:
15487:
15481:
15474:
15473:External links
15471:
15469:
15468:
15446:
15428:
15418:(2): 155–163.
15400:
15394:
15377:
15375:
15372:
15370:
15369:
15322:
15301:
15278:
15263:
15245:
15225:
15219:Physica Medica
15210:
15173:(50): 505202.
15157:
15118:
15057:
15007:
14988:(1): 128–138.
14972:
14965:
14939:
14890:
14870:
14854:
14806:
14788:
14753:
14746:
14728:
14721:
14703:
14696:
14673:
14661:
14630:
14599:
14575:M.A. Sanders.
14566:
14564:
14561:
14559:
14558:
14541:
14537:
14534:
14527:
14524:
14521:
14518:
14515:
14512:
14511:
14507:
14501:
14498:
14493:
14490:
14486:
14482:
14481:
14477:
14471:
14467:
14461:
14454:
14451:
14448:
14445:
14442:
14439:
14436:
14411:
14404:
14400:
14395:
14390:
14387:
14381:
14376:
14371:
14368:
14365:
14362:
14357:
14353:
14349:
14346:
14343:
14340:
14337:
14332:
14329:
14326:
14322:
14316:
14312:
14308:
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14300:
14294:
14291:
14288:
14285:
14282:
14262:
14259:
14256:
14253:
14250:
14236:
14231:
14226:
14221:
14216:
14211:
14206:
14201:
14195:
14194:
14193:
14177:
14174:
14111:
14108:
14065:
14064:
14061:
14058:
14055:
14052:
14049:
14046:
14043:
14040:
14037:
14034:
14031:
14024:
14023:
14020:
14017:
14014:
14011:
14008:
14005:
14002:
13999:
13996:
13993:
13990:
13986:
13985:
13982:
13979:
13976:
13973:
13970:
13967:
13964:
13961:
13958:
13955:
13952:
13948:
13947:
13944:
13941:
13938:
13935:
13932:
13929:
13926:
13923:
13920:
13917:
13914:
13910:
13909:
13906:
13903:
13900:
13897:
13894:
13891:
13888:
13885:
13882:
13879:
13876:
13872:
13871:
13868:
13865:
13862:
13859:
13856:
13853:
13850:
13847:
13844:
13841:
13838:
13834:
13833:
13830:
13827:
13824:
13821:
13818:
13815:
13812:
13809:
13806:
13803:
13800:
13796:
13795:
13792:
13789:
13786:
13783:
13780:
13777:
13774:
13771:
13768:
13765:
13762:
13758:
13757:
13754:
13751:
13748:
13745:
13742:
13739:
13736:
13733:
13730:
13727:
13724:
13720:
13719:
13716:
13713:
13710:
13707:
13704:
13701:
13698:
13695:
13692:
13689:
13686:
13682:
13681:
13678:
13675:
13672:
13669:
13666:
13663:
13660:
13657:
13654:
13651:
13648:
13644:
13643:
13629:
13625:
13614:
13594:
13590:
13565:-value. A low
13536:{\textstyle p}
13532:
13519:
13502:
13500:
13497:
13487:
13486:
13471:
13466:
13459:
13455:
13449:
13445:
13439:
13432:
13427:
13424:
13421:
13417:
13404:
13398:
13397:
13382:
13377:
13370:
13366:
13359:
13355:
13351:
13346:
13342:
13335:
13328:
13323:
13320:
13317:
13313:
13300:
13294:
13293:
13280:
13275:
13268:
13264:
13258:
13254:
13248:
13241:
13236:
13233:
13230:
13226:
13215:
13209:
13208:
13195:
13190:
13183:
13179:
13172:
13168:
13164:
13159:
13155:
13148:
13141:
13136:
13133:
13130:
13126:
13115:
13111:
13110:
13107:
13101:
13100:
13088:
13085:
13082:
13079:
13076:
13073:
13070:
13067:
13064:
13059:
13054:
13050:
13046:
13041:
13037:
13033:
13030:
13027:
13022:
13018:
13002:
12988:
12984:
12978:
12973:
12970:
12967:
12963:
12957:
12954:
12949:
12944:
12939:
12935:
12910:
12905:
12902:
12899:
12895:
12889:
12885:
12881:
12876:
12872:
12866:
12861:
12857:
12851:
12846:
12842:
12838:
12833:
12828:
12825:
12822:
12818:
12794:
12789:
12785:
12781:
12778:
12775:
12772:
12747:
12743:
12739:
12736:
12733:
12730:
12727:
12722:
12718:
12695:F-distribution
12666:
12663:
12650:
12646:
12642:
12622:
12618:
12614:
12594:
12574:
12552:
12547:
12543:
12539:
12536:
12504:
12499:
12496:
12491:
12487:
12484:
12481:
12478:
12456:
12451:
12447:
12443:
12440:
12408:
12404:
12401:
12396:
12393:
12387:
12383:
12380:
12377:
12356:
12350:
12347:
12342:
12337:
12334:
12328:
12324:
12321:
12318:
12292:
12287:
12283:
12279:
12276:
12264:
12261:
12241:Main article:
12238:
12235:
12223:Main article:
12220:
12217:
12215:
12212:
12189:
12185:
12179:
12175:
12169:
12164:
12161:
12158:
12154:
12150:
12147:
12125:
12122:
12117:
12112:
12107:
12103:
12099:
12096:
12093:
12088:
12084:
12061:
12057:
12053:
12050:
12047:
12042:
12038:
12025:
12022:
12012:
12009:
12008:
12007:
11993:
11989:
11985:
11980:
11976:
11953:
11949:
11926:
11922:
11899:
11892:
11888:
11884:
11879:
11875:
11870:
11866:
11861:
11857:
11853:
11848:
11844:
11821:
11814:
11810:
11805:
11801:
11796:
11792:
11769:
11762:
11758:
11753:
11749:
11744:
11740:
11728:
11714:
11707:
11703:
11698:
11694:
11689:
11685:
11662:
11655:
11651:
11646:
11642:
11637:
11633:
11606:
11602:
11596:
11589:
11585:
11575:
11571:
11565:
11558:
11554:
11546:
11543:
11523:
11518:
11514:
11510:
11505:
11501:
11497:
11494:
11491:
11488:
11464:
11450:
11449:
11438:
11433:
11428:
11424:
11420:
11412:
11407:
11399:
11395:
11389:
11385:
11379:
11376:
11373:
11366:
11362:
11356:
11352:
11345:
11340:
11336:
11328:
11324:
11318:
11314:
11308:
11305:
11302:
11295:
11291:
11285:
11281:
11274:
11269:
11244:
11241:
11238:
11235:
11232:
11229:
11226:
11223:
11220:
11217:
11214:
11209:
11205:
11184:
11181:
11176:
11172:
11168:
11165:
11162:
11157:
11153:
11132:
11112:
11092:
11072:
11069:
11066:
11063:
11060:
11057:
11054:
11051:
11031:
11028:
11025:
11005:
10982:
10979:
10976:
10956:
10953:
10948:
10944:
10933:quadratic form
10920:
10917:
10914:
10884:
10881:
10878:
10858:
10835:
10815:
10777:
10757:
10754:
10751:
10748:
10745:
10740:
10737:
10733:
10727:
10723:
10719:
10716:
10713:
10710:
10707:
10704:
10684:
10664:
10644:
10624:
10604:
10577:
10557:
10546:
10545:
10539:
10530:
10521:
10515:
10509:
10495:
10490:
10486:
10482:
10479:
10475:
10471:
10466:
10460:
10455:
10450:
10447:
10442:
10438:
10433:
10429:
10421:
10416:
10413:
10410:
10406:
10385:
10382:
10379:
10376:
10373:
10347:
10343:
10331:
10317:
10312:
10309:
10305:
10301:
10296:
10291:
10287:
10284:
10279:
10275:
10270:
10266:
10258:
10253:
10250:
10247:
10243:
10221:
10218:
10215:
10212:
10209:
10206:
10203:
10200:
10195:
10191:
10179:
10165:
10160:
10156:
10152:
10149:
10146:
10143:
10140:
10137:
10134:
10131:
10106:
10103:
10100:
10097:
10094:
10091:
10088:
10085:
10082:
10071:
10054:
10048:
10043:
10039:
10032:
10026:
10021:
10017:
10010:
10007:
10004:
10001:
9994:
9991:
9988:
9984:
9958:
9951:
9947:
9942:
9938:
9935:
9912:
9905:
9901:
9896:
9892:
9889:
9878:
9872:
9853:
9848:
9844:
9840:
9833:
9830:
9827:
9823:
9817:
9814:
9789:
9784:
9780:
9776:
9773:
9762:
9748:
9743:
9739:
9735:
9730:
9726:
9700:
9697:
9694:
9691:
9688:
9685:
9682:
9671:
9657:
9652:
9648:
9644:
9639:
9635:
9609:
9606:
9603:
9600:
9597:
9594:
9591:
9580:
9567:
9562:
9559:
9555:
9551:
9548:
9545:
9542:
9522:
9519:
9516:
9513:
9510:
9507:
9504:
9501:
9498:
9487:
9471:
9468:
9464:
9460:
9457:
9454:
9451:
9448:
9445:
9442:
9439:
9417:
9412:
9409:
9405:
9401:
9398:
9387:
9367:
9364:
9360:
9356:
9353:
9350:
9347:
9344:
9341:
9319:
9314:
9310:
9306:
9303:
9292:
9274:
9270:
9266:
9261:
9237:
9232:
9228:
9224:
9221:
9210:
9193:
9190:
9187:
9184:
9181:
9178:
9175:
9171:
9167:
9164:
9161:
9158:
9155:
9152:
9149:
9146:
9125:
9122:
9119:
9096:
9091:
9087:
9083:
9080:
9069:
9040:
9035:
9031:
9027:
9022:
9018:
9014:
9011:
9008:
9005:
9002:
8997:
8994:
8991:
8988:
8985:
8982:
8979:
8974:
8969:
8958:
8957:
8956:
8955:
8942:
8937:
8933:
8911:
8906:
8903:
8898:
8894:
8889:
8885:
8882:
8861:
8856:
8852:
8848:
8845:
8842:
8838:
8834:
8831:
8817:
8816:
8803:
8796:
8792:
8787:
8766:
8761:
8757:
8751:
8748:
8743:
8739:
8734:
8730:
8727:
8707:
8702:
8698:
8694:
8689:
8685:
8681:
8677:
8673:
8670:
8659:
8647:
8644:
8641:
8617:
8614:
8611:
8606:
8601:
8595:
8592:
8586:
8581:
8576:
8572:
8561:
8544:
8541:
8538:
8535:
8532:
8529:
8520:
8516:
8506:
8503:
8497:
8493:
8490:
8487:
8482:
8477:
8473:
8469:
8449:
8446:
8443:
8429:
8428:
8411:September 2011
8387:
8385:
8378:
8372:
8369:
8365:
8364:
8363:
8362:
8347:
8341:
8332:
8329:
8325:
8320:
8317:
8312:
8307:
8276:
8270:
8267:
8263:
8238:
8235:
8231:
8226:
8223:
8200:
8195:
8191:
8187:
8164:
8161:
8158:
8153:
8149:
8145:
8142:
8131:
8113:
8110:
8107:
8104:
8080:
8077:
8055:
8052:
8049:
8044:
8040:
8036:
8033:
8000:
7996:
7975:
7970:
7966:
7962:
7959:
7956:
7933:
7929:
7925:
7899:
7895:
7891:
7860:
7857:
7851:
7847:
7844:
7841:
7838:
7835:
7815:
7795:
7792:
7789:
7784:
7780:
7776:
7773:
7749:
7746:
7743:
7723:
7703:
7671:
7668:
7655:
7652:
7648:
7644:
7641:
7638:
7635:
7615:
7591:
7587:
7583:
7579:
7575:
7572:
7569:
7566:
7563:
7558:
7554:
7550:
7547:
7542:
7539:
7536:
7532:
7528:
7524:
7520:
7517:
7514:
7511:
7506:
7503:
7500:
7496:
7492:
7488:
7484:
7481:
7478:
7475:
7472:
7467:
7463:
7459:
7456:
7453:
7450:
7447:
7425:
7422:
7419:
7415:
7411:
7407:
7386:
7366:
7344:
7339:
7315:
7311:
7307:
7304:
7301:
7298:
7295:
7292:
7289:
7278:
7277:
7266:
7263:
7260:
7257:
7254:
7251:
7248:
7245:
7240:
7237:
7232:
7229:
7226:
7223:
7220:
7217:
7214:
7211:
7201:
7190:
7187:
7184:
7181:
7178:
7175:
7172:
7169:
7166:
7163:
7160:
7155:
7152:
7147:
7144:
7141:
7138:
7135:
7132:
7129:
7126:
7111:
7108:
7107:
7106:
7095:
7091:
7088:
7085:
7082:
7079:
7076:
7071:
7068:
7065:
7061:
7057:
7052:
7048:
7024:
7021:
7020:
7019:
7008:
7001:
6996:
6993:
6988:
6984:
6978:
6972:
6969:
6964:
6961:
6957:
6953:
6945:
6941:
6937:
6934:
6931:
6928:
6925:
6922:
6919:
6916:
6913:
6910:
6907:
6904:
6901:
6898:
6895:
6892:
6889:
6886:
6883:
6880:
6877:
6874:
6871:
6866:
6862:
6858:
6855:
6852:
6829:
6817:
6814:
6793:
6790:
6787:
6784:
6781:
6778:
6775:
6772:
6768:
6764:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6740:
6737:
6734:
6731:
6728:
6708:
6705:
6702:
6699:
6696:
6693:
6690:
6670:
6639:
6636:
6633:
6630:
6619:
6618:
6607:
6603:
6598:
6595:
6590:
6585:
6580:
6574:
6571:
6566:
6563:
6559:
6555:
6551:
6546:
6541:
6538:
6533:
6529:
6525:
6521:
6517:
6514:
6511:
6506:
6503:
6498:
6495:
6492:
6488:
6485:
6481:
6478:
6475:
6472:
6469:
6466:
6463:
6460:
6456:
6453:
6450:
6447:
6442:
6437:
6433:
6429:
6426:
6407:
6404:
6389:
6385:
6382:
6376:
6371:
6367:
6344:
6341:
6319:
6315:
6295:
6275:
6249:
6246:
6242:
6238:
6234:
6231:
6226:
6222:
6218:
6215:
6212:
6209:
6206:
6203:
6198:
6195:
6192:
6188:
6182:
6179:
6153:
6149:
6144:
6141:
6136:
6132:
6111:
6108:
6105:
6102:
6099:
6079:
6068:Asymptotically
6065:
6064:
6053:
6050:
6047:
6042:
6038:
6034:
6029:
6025:
6014:
6010:
6006:
6002:
5999:
5996:
5993:
5990:
5986:
5982:
5978:
5975:
5972:
5968:
5964:
5961:
5958:
5953:
5949:
5943:
5938:
5935:
5932:
5928:
5922:
5919:
5914:
5909:
5906:
5881:
5861:
5841:
5818:
5806:
5803:
5788:
5784:
5780:
5777:
5774:
5769:
5765:
5742:
5738:
5734:
5731:
5728:
5723:
5719:
5715:
5712:
5690:
5686:
5683:
5680:
5674:
5671:
5649:
5645:
5622:
5618:
5615:
5612:
5606:
5603:
5600:
5595:
5591:
5578:
5575:
5560:
5555:
5550:
5547:
5544:
5540:
5533:
5525:
5520:
5516:
5512:
5509:
5506:
5503:
5500:
5495:
5490:
5486:
5479:
5473:
5470:
5467:
5461:
5457:
5450:
5446:
5439:
5433:
5430:
5424:
5420:
5413:
5405:
5401:
5394:
5391:
5385:
5380:
5376:
5372:
5367:
5362:
5359:
5356:
5352:
5331:
5328:
5323:
5320:
5314:
5311:
5305:
5300:
5295:
5292:
5289:
5286:
5281:
5275:
5271:
5267:
5261:
5258:
5252:
5247:
5242:
5239:
5233:
5229:
5225:
5222:
5202:
5197:
5193:
5189:
5186:
5183:
5180:
5177:
5172:
5168:
5164:
5161:
5158:
5138:
5116:
5112:
5089:
5085:
5081:
5078:
5075:
5072:
5069:
5064:
5060:
5039:
5036:
5033:
5013:
4990:
4987:
4976:
4972:
4965:
4960:
4956:
4935:
4912:
4909:
4886:
4881:
4861:
4858:
4852:
4848:
4841:
4835:
4832:
4827:
4820:
4817:
4807:
4804:
4794:
4791:
4784:
4781:
4776:
4773:
4768:
4764:
4757:
4749:
4742:
4739:
4732:
4729:
4724:
4719:
4715:
4709:
4704:
4701:
4698:
4694:
4687:
4679:
4675:
4668:
4665:
4659:
4654:
4650:
4646:
4641:
4636:
4633:
4630:
4626:
4602:
4599:
4576:
4556:
4553:
4548:
4545:
4539:
4536:
4530:
4525:
4520:
4517:
4503:
4502:
4490:
4485:
4481:
4475:
4470:
4467:
4464:
4460:
4454:
4451:
4446:
4440:
4437:
4412:
4407:
4404:
4401:
4397:
4393:
4388:
4384:
4377:
4374:
4368:
4363:
4359:
4355:
4350:
4345:
4342:
4339:
4335:
4304:
4300:
4296:
4293:
4290:
4287:
4284:
4279:
4275:
4253:Main article:
4250:
4247:
4245:
4242:
4230:
4229:
4218:
4213:
4209:
4205:
4201:
4195:
4192:
4189:
4185:
4181:
4178:
4175:
4172:
4169:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4124:
4121:
4118:
4096:
4091:
4087:
4083:
4079:
4073:
4070:
4067:
4063:
4059:
4056:
4053:
4050:
4047:
4043:
4040:
4037:
4034:
4031:
4011:
4008:
4005:
4002:
3999:
3975:
3971:
3967:
3964:
3961:
3927:
3907:
3904:
3900:
3897:
3894:
3891:
3869:
3865:
3861:
3858:
3854:
3848:
3845:
3840:
3837:
3834:
3830:
3827:
3824:
3821:
3810:
3809:
3796:
3792:
3788:
3785:
3781:
3777:
3774:
3771:
3768:
3765:
3761:
3758:
3755:
3752:
3729:
3726:
3723:
3696:
3693:
3690:
3687:
3684:
3681:
3657:
3654:
3651:
3648:
3645:
3642:
3631:
3630:
3619:
3615:
3609:
3606:
3600:
3595:
3592:
3586:
3582:
3579:
3573:
3568:
3565:
3560:
3557:
3552:
3547:
3544:
3538:
3533:
3530:
3525:
3522:
3516:
3513:
3510:
3506:
3503:
3500:
3497:
3466:
3463:
3460:
3440:
3437:
3420:
3396:
3383:gamma function
3370:
3367:
3363:
3359:
3356:
3353:
3342:
3341:
3328:
3323:
3315:
3313:
3310:
3307:
3306:
3303:
3300:
3297:
3294:
3291:
3289:
3281:
3276:
3273:
3268:
3264:
3259:
3255:
3251:
3247:
3239:
3235:
3231:
3228:
3224:
3218:
3215:
3212:
3208:
3204:
3200:
3192:
3191:
3189:
3184:
3181:
3178:
3174:
3171:
3168:
3165:
3146:
3143:
3130:
3107:
3085:
3081:
3060:
3040:
3018:
3014:
3010:
3007:
3002:
2998:
2977:
2955:
2951:
2941:distribution;
2928:
2924:
2901:
2897:
2883:
2882:
2867:
2863:
2856:
2852:
2846:
2842:
2838:
2833:
2829:
2825:
2817:
2812:
2809:
2806:
2802:
2798:
2793:
2789:
2770:
2769:
2755:
2752:
2745:
2741:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2713:
2707:
2704:
2697:
2693:
2689:
2686:
2683:
2680:
2677:
2671:
2666:
2662:
2638:
2635:
2632:
2629:
2626:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2521:
2520:
2506:
2503:
2500:
2493:
2489:
2485:
2482:
2479:
2476:
2473:
2467:
2462:
2458:
2431:
2428:
2425:
2422:
2419:
2399:
2379:
2359:
2348:
2347:
2332:
2329:
2326:
2320:
2317:
2314:
2311:
2305:
2302:
2249:
2244:
2240:
2233:
2227:
2204:
2182:
2178:
2174:
2171:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2110:
2090:
2070:
2007:
2006:
2001:
1996:
1990:
1984:
1978:
1972:
1949:
1946:
1939:
1925:
1924:
1913:
1908:
1903:
1899:
1892:
1886:
1866:
1863:
1860:
1855:
1851:
1844:
1838:
1820:
1819:
1808:
1803:
1798:
1794:
1788:
1783:
1780:
1777:
1773:
1769:
1763:
1737:
1730:
1722:
1719:
1652:
1649:
1646:
1641:
1637:
1633:
1628:
1618:
1615:
1595:
1590:
1586:
1582:
1579:
1576:
1573:
1568:
1565:
1560:
1557:
1554:
1546:
1543:
1521:
1516:
1512:
1506:
1502:
1498:
1495:
1465:
1460:
1456:
1450:
1446:
1419:
1416:
1413:
1410:
1407:
1402:
1392:
1389:
1369:
1349:
1329:
1326:
1323:
1320:
1317:
1312:
1309:
1304:
1301:
1298:
1290:
1287:
1265:
1260:
1256:
1252:
1249:
1219:
1214:
1210:
1180:
1157:
1133:
1129:
1098:
1097:
1083:
1078:
1075:
1072:
1069:
1059:
1055:
1051:
1048:
1044:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1012:
1006:
1005:
992:
988:
984:
981:
977:
973:
970:
967:
964:
961:
958:
948:
942:
941:
927:
924:
919:
916:
906:
902:
898:
895:
891:
887:
884:
881:
878:
875:
865:
859:
858:
842:
837:
834:
829:
825:
821:
815:
812:
807:
804:
800:
796:
792:
790:
788:
784:
778:
771:
768:
761:
756:
753:
749:
745:
742:
739:
736:
734:
730:
727:
722:
721:
711:
705:
704:
691:
688:
676:
670:
669:
655:
651:
647:
635:
629:
628:
616:
613:
603:
597:
596:
584:
581:
578:
575:
572:
569:
566:
563:
553:
547:
546:
532:
526:
517:
514:
510:
505:
502:
497:
492:
489:
479:
473:
472:
461:
451:
445:
444:
431:
425:
422:
416:
411:
408:
402:
398:
391:
388:
384:
380:
377:
374:
370:
358:
352:
351:
337:
333:
329:
326:
322:
316:
313:
310:
306:
302:
298:
290:
287:
283:
279:
276:
273:
268:
264:
260:
256:
251:
239:
233:
232:
220:
217:
214:
211:
208:
205:
202:
199:
189:
183:
182:
162:
157:
152:
149:
139:
133:
132:
118:
113:
109:
87:
84:
81:
76:
72:
61:
57:
56:
48:
45:
44:
36:
15:
9:
6:
4:
3:
2:
16927:
16916:
16913:
16911:
16908:
16907:
16905:
16890:
16882:
16880:
16872:
16871:
16868:
16862:
16859:
16857:
16854:
16852:
16849:
16847:
16844:
16842:
16839:
16837:
16834:
16832:
16829:
16827:
16824:
16822:
16819:
16817:
16814:
16812:
16809:
16808:
16806:
16802:
16796:
16793:
16790:
16786:
16784:
16781:
16778:
16774:
16773:
16771:
16769:
16764:
16760:
16754:
16751:
16749:
16746:
16743:
16739:
16737:
16734:
16731:
16727:
16725:
16722:
16719:
16715:
16713:
16710:
16708:
16705:
16703:
16700:
16698:
16695:
16693:
16690:
16688:
16685:
16683:
16680:
16677:
16676:
16670:
16669:
16667:
16665:
16661:
16653:
16650:
16648:
16645:
16643:
16640:
16638:
16635:
16634:
16633:
16630:
16626:
16623:
16622:
16621:
16618:
16616:
16615:
16610:
16608:
16607:Matrix normal
16605:
16603:
16600:
16597:
16596:
16591:
16587:
16584:
16583:
16582:
16579:
16577:
16576:
16573:Multivariate
16571:
16569:
16566:
16564:
16561:
16559:
16556:
16552:
16549:
16548:
16547:
16544:
16541:
16537:
16533:
16530:
16528:
16525:
16524:
16523:
16520:
16518:
16515:
16512:
16508:
16507:
16505:
16503:
16500:Multivariate
16497:
16487:
16484:
16483:
16481:
16475:
16472:
16466:
16456:
16453:
16451:
16448:
16446:
16444:
16440:
16438:
16436:
16432:
16430:
16428:
16424:
16422:
16420:
16415:
16413:
16411:
16406:
16404:
16402:
16397:
16395:
16393:
16388:
16386:
16384:
16379:
16377:
16374:
16372:
16369:
16367:
16364:
16362:
16359:
16358:
16356:
16352:with support
16350:
16344:
16341:
16339:
16336:
16334:
16331:
16329:
16328:
16323:
16321:
16318:
16316:
16313:
16311:
16308:
16306:
16303:
16301:
16298:
16296:
16295:
16290:
16288:
16285:
16281:
16278:
16277:
16276:
16273:
16271:
16268:
16266:
16265:
16257:
16255:
16252:
16250:
16247:
16245:
16242:
16240:
16237:
16235:
16232:
16230:
16227:
16225:
16224:
16219:
16217:
16214:
16212:
16211:
16206:
16204:
16201:
16199:
16196:
16195:
16193:
16189:on the whole
16185:
16179:
16176:
16172:
16169:
16168:
16167:
16164:
16162:
16161:type-2 Gumbel
16159:
16157:
16154:
16152:
16149:
16147:
16144:
16142:
16139:
16137:
16134:
16132:
16129:
16127:
16124:
16122:
16119:
16117:
16114:
16112:
16109:
16107:
16104:
16102:
16099:
16097:
16094:
16092:
16089:
16087:
16084:
16082:
16079:
16077:
16074:
16072:
16069:
16067:
16064:
16062:
16059:
16055:
16052:
16051:
16050:
16047:
16045:
16043:
16038:
16036:
16033:
16031:
16030:Half-logistic
16028:
16024:
16021:
16020:
16019:
16016:
16014:
16011:
16007:
16004:
16002:
15999:
15998:
15997:
15994:
15992:
15989:
15987:
15986:Folded normal
15984:
15980:
15977:
15976:
15975:
15974:
15970:
15966:
15963:
15961:
15958:
15956:
15953:
15952:
15951:
15948:
15944:
15941:
15940:
15939:
15936:
15934:
15931:
15929:
15926:
15920:
15917:
15916:
15915:
15912:
15910:
15907:
15906:
15905:
15902:
15900:
15897:
15895:
15892:
15890:
15887:
15885:
15882:
15880:
15877:
15875:
15872:
15871:
15869:
15861:
15855:
15852:
15850:
15847:
15845:
15842:
15840:
15837:
15835:
15832:
15830:
15829:Raised cosine
15827:
15825:
15822:
15820:
15817:
15815:
15812:
15810:
15807:
15805:
15802:
15800:
15797:
15795:
15792:
15790:
15787:
15785:
15782:
15780:
15777:
15775:
15772:
15770:
15767:
15766:
15764:
15758:
15755:
15749:
15739:
15736:
15734:
15731:
15729:
15726:
15724:
15721:
15719:
15716:
15714:
15711:
15709:
15706:
15704:
15703:Mixed Poisson
15701:
15699:
15696:
15694:
15691:
15689:
15686:
15684:
15681:
15679:
15676:
15674:
15671:
15669:
15666:
15664:
15661:
15659:
15656:
15654:
15651:
15650:
15648:
15642:
15636:
15633:
15631:
15628:
15626:
15623:
15621:
15618:
15616:
15613:
15611:
15608:
15604:
15601:
15600:
15599:
15596:
15594:
15591:
15589:
15586:
15584:
15583:Beta-binomial
15581:
15579:
15576:
15574:
15571:
15570:
15568:
15562:
15559:
15553:
15548:
15544:
15537:
15532:
15530:
15525:
15523:
15518:
15517:
15514:
15508:
15505:
15503:
15500:
15498:
15496:
15492:
15488:
15485:
15482:
15480:
15477:
15476:
15465:
15461:
15457:
15453:
15447:
15443:
15439:
15438:
15433:
15429:
15425:
15421:
15417:
15413:
15409:
15405:
15401:
15397:
15391:
15387:
15383:
15379:
15378:
15365:
15361:
15357:
15353:
15349:
15345:
15341:
15337:
15333:
15326:
15319:
15315:
15311:
15305:
15298:
15295:
15291:
15287:
15286:F. R. Helmert
15282:
15275:
15270:
15268:
15259:
15255:
15249:
15241:
15237:
15234:
15229:
15223:
15220:
15214:
15206:
15202:
15198:
15194:
15190:
15186:
15181:
15176:
15172:
15168:
15161:
15153:
15149:
15145:
15141:
15137:
15133:
15129:
15122:
15114:
15110:
15105:
15100:
15095:
15090:
15086:
15082:
15078:
15074:
15073:
15068:
15061:
15053:
15049:
15045:
15041:
15036:
15031:
15027:
15023:
15022:
15014:
15012:
15003:
14999:
14995:
14991:
14987:
14983:
14976:
14968:
14962:
14958:
14953:
14952:
14943:
14935:
14931:
14927:
14923:
14918:
14913:
14909:
14905:
14901:
14894:
14888:
14884:
14880:
14877:M. K. Simon,
14874:
14867:
14863:
14858:
14844:
14840:
14836:
14832:
14828:
14824:
14817:
14810:
14802:
14795:
14793:
14784:
14780:
14776:
14772:
14769:(2): 173–82.
14768:
14764:
14757:
14749:
14743:
14739:
14732:
14724:
14718:
14714:
14707:
14699:
14693:
14689:
14682:
14680:
14678:
14671:
14668:NIST (2006).
14665:
14657:
14653:
14649:
14645:
14641:
14637:
14633:
14627:
14623:
14622:
14617:
14613:
14609:
14603:
14589:on 2011-07-15
14585:
14578:
14571:
14567:
14556:
14539:
14535:
14532:
14522:
14519:
14516:
14505:
14499:
14496:
14491:
14488:
14484:
14475:
14469:
14459:
14452:
14446:
14443:
14440:
14409:
14402:
14398:
14393:
14388:
14385:
14379:
14366:
14363:
14360:
14355:
14351:
14347:
14344:
14338:
14335:
14330:
14327:
14324:
14320:
14314:
14310:
14306:
14302:
14298:
14292:
14286:
14280:
14254:
14251:
14240:
14237:
14235:
14232:
14230:
14227:
14225:
14222:
14220:
14217:
14215:
14212:
14210:
14207:
14205:
14202:
14200:
14197:
14196:
14191:
14185:
14180:
14173:
14171:
14168:(Σ being the
14166:
14162:
14152:
14148:
14144:
14140:
14139:Elderton 1902
14136:
14132:
14128:
14123:
14121:
14117:
14107:
14105:
14103:
14097:
14090:2.1673 ≈ 2.17
14081:
14072:
14062:
14059:
14056:
14053:
14050:
14047:
14044:
14041:
14038:
14035:
14032:
14029:
14025:
14021:
14018:
14015:
14012:
14009:
14006:
14003:
14000:
13997:
13994:
13991:
13988:
13987:
13983:
13980:
13977:
13974:
13971:
13968:
13965:
13962:
13959:
13956:
13953:
13950:
13949:
13945:
13942:
13939:
13936:
13933:
13930:
13927:
13924:
13921:
13918:
13915:
13912:
13911:
13907:
13904:
13901:
13898:
13895:
13892:
13889:
13886:
13883:
13880:
13877:
13874:
13873:
13869:
13866:
13863:
13860:
13857:
13854:
13851:
13848:
13845:
13842:
13839:
13836:
13835:
13831:
13828:
13825:
13822:
13819:
13816:
13813:
13810:
13807:
13804:
13801:
13798:
13797:
13793:
13790:
13787:
13784:
13781:
13778:
13775:
13772:
13769:
13766:
13763:
13760:
13759:
13755:
13752:
13749:
13746:
13743:
13740:
13737:
13734:
13731:
13728:
13725:
13722:
13721:
13717:
13714:
13711:
13708:
13705:
13702:
13699:
13696:
13693:
13690:
13687:
13684:
13683:
13679:
13676:
13673:
13670:
13667:
13664:
13661:
13658:
13655:
13652:
13649:
13646:
13645:
13627:
13623:
13612:
13611:
13608:
13592:
13588:
13579:
13574:
13572:
13568:
13564:
13560:
13556:
13552:
13548:
13544:
13530:
13516:
13515:
13508:
13496:
13494:
13469:
13464:
13457:
13453:
13447:
13443:
13437:
13430:
13425:
13422:
13419:
13415:
13405:
13403:
13400:
13399:
13380:
13375:
13368:
13364:
13357:
13353:
13349:
13344:
13340:
13333:
13326:
13321:
13318:
13315:
13311:
13301:
13299:
13296:
13295:
13278:
13273:
13266:
13262:
13256:
13252:
13246:
13239:
13234:
13231:
13228:
13224:
13216:
13214:
13211:
13210:
13193:
13188:
13181:
13177:
13170:
13166:
13162:
13157:
13153:
13146:
13139:
13134:
13131:
13128:
13124:
13116:
13113:
13112:
13108:
13105:
13104:
13086:
13083:
13080:
13077:
13074:
13071:
13068:
13065:
13057:
13052:
13048:
13044:
13039:
13035:
13028:
13025:
13020:
13016:
13007:
13003:
12986:
12982:
12976:
12971:
12968:
12965:
12961:
12955:
12952:
12947:
12937:
12933:
12908:
12903:
12900:
12897:
12893:
12887:
12883:
12879:
12874:
12859:
12855:
12849:
12844:
12840:
12831:
12826:
12823:
12820:
12816:
12807:
12787:
12783:
12779:
12776:
12770:
12763:
12745:
12741:
12737:
12734:
12731:
12728:
12725:
12720:
12716:
12707:
12706:
12705:
12702:
12700:
12696:
12692:
12688:
12684:
12680:
12676:
12672:
12662:
12648:
12644:
12640:
12620:
12616:
12612:
12592:
12572:
12550:
12545:
12541:
12537:
12534:
12526:
12521:
12519:
12502:
12497:
12494:
12489:
12485:
12482:
12479:
12476:
12454:
12449:
12445:
12441:
12438:
12430:
12425:
12406:
12402:
12399:
12394:
12391:
12385:
12378:
12375:
12354:
12348:
12345:
12340:
12335:
12332:
12326:
12319:
12316:
12308:
12290:
12285:
12281:
12277:
12274:
12260:
12244:
12234:
12232:
12226:
12211:
12209:
12205:
12187:
12183:
12177:
12173:
12167:
12162:
12159:
12156:
12152:
12148:
12145:
12123:
12120:
12110:
12105:
12101:
12097:
12094:
12091:
12086:
12082:
12059:
12055:
12051:
12048:
12045:
12040:
12036:
12021:
11991:
11987:
11983:
11978:
11974:
11951:
11947:
11924:
11920:
11897:
11890:
11886:
11882:
11877:
11873:
11868:
11864:
11859:
11855:
11851:
11846:
11842:
11819:
11812:
11808:
11803:
11799:
11794:
11790:
11767:
11760:
11756:
11751:
11747:
11742:
11738:
11729:
11712:
11705:
11701:
11696:
11692:
11687:
11683:
11660:
11653:
11649:
11644:
11640:
11635:
11631:
11604:
11600:
11594:
11587:
11583:
11573:
11569:
11563:
11556:
11552:
11544:
11541:
11516:
11512:
11508:
11503:
11499:
11492:
11489:
11486:
11478:
11477:F-distributed
11462:
11455:
11454:
11453:
11436:
11431:
11426:
11422:
11418:
11405:
11397:
11393:
11387:
11383:
11377:
11374:
11371:
11364:
11360:
11354:
11350:
11343:
11334:
11326:
11322:
11316:
11312:
11306:
11303:
11300:
11293:
11289:
11283:
11279:
11272:
11267:
11258:
11257:
11256:
11242:
11239:
11236:
11233:
11230:
11227:
11224:
11221:
11218:
11215:
11212:
11207:
11203:
11182:
11179:
11174:
11170:
11166:
11163:
11160:
11155:
11151:
11130:
11110:
11090:
11064:
11061:
11055:
11052:
11049:
11029:
11026:
11023:
10994:
10980:
10977:
10974:
10954:
10951:
10946:
10942:
10934:
10918:
10915:
10912:
10905:
10901:
10897:
10882:
10879:
10876:
10856:
10848:
10833:
10813:
10804:
10802:
10798:
10794:
10789:
10775:
10752:
10749:
10746:
10738:
10735:
10731:
10725:
10717:
10714:
10711:
10705:
10702:
10682:
10662:
10642:
10622:
10602:
10593:
10591:
10575:
10555:
10543:
10540:
10538:
10534:
10531:
10529:
10525:
10522:
10519:
10516:
10514:
10510:
10493:
10488:
10484:
10480:
10477:
10473:
10469:
10464:
10458:
10448:
10445:
10440:
10436:
10427:
10419:
10414:
10411:
10408:
10404:
10383:
10380:
10377:
10374:
10371:
10363:
10345:
10341:
10332:
10315:
10310:
10307:
10303:
10299:
10294:
10285:
10282:
10277:
10273:
10264:
10256:
10251:
10248:
10245:
10241:
10216:
10213:
10210:
10204:
10201:
10198:
10193:
10189:
10180:
10163:
10158:
10154:
10150:
10144:
10138:
10135:
10132:
10129:
10121:
10101:
10098:
10095:
10089:
10083:
10080:
10072:
10069:
10046:
10041:
10037:
10030:
10024:
10019:
10015:
10005:
10002:
9999:
9992:
9989:
9986:
9982:
9956:
9949:
9945:
9940:
9936:
9933:
9910:
9903:
9899:
9894:
9890:
9887:
9879:
9877:
9873:
9870:
9851:
9846:
9842:
9838:
9821:
9815:
9812:
9787:
9782:
9778:
9774:
9771:
9763:
9746:
9741:
9737:
9733:
9728:
9724:
9715:
9695:
9689:
9686:
9683:
9680:
9672:
9655:
9650:
9646:
9642:
9637:
9633:
9624:
9604:
9598:
9595:
9592:
9589:
9581:
9565:
9560:
9557:
9553:
9549:
9546:
9543:
9540:
9517:
9514:
9511:
9505:
9502:
9499:
9496:
9488:
9485:
9466:
9462:
9458:
9455:
9452:
9446:
9443:
9440:
9437:
9415:
9410:
9407:
9403:
9399:
9396:
9388:
9385:
9381:
9362:
9358:
9354:
9348:
9345:
9342:
9339:
9317:
9312:
9308:
9304:
9301:
9293:
9290:
9272:
9268:
9264:
9259:
9235:
9230:
9226:
9222:
9219:
9211:
9208:
9188:
9185:
9182:
9179:
9176:
9173:
9169:
9165:
9162:
9159:
9150:
9147:
9144:
9123:
9120:
9117:
9094:
9089:
9085:
9081:
9078:
9070:
9067:
9064:
9060:
9056:
9053:(The squared
9038:
9033:
9029:
9025:
9020:
9009:
9006:
9003:
8995:
8992:
8989:
8986:
8983:
8980:
8977:
8960:
8959:
8940:
8935:
8931:
8909:
8896:
8892:
8883:
8880:
8854:
8850:
8846:
8843:
8832:
8829:
8821:
8820:
8819:
8818:
8801:
8794:
8790:
8785:
8764:
8759:
8755:
8741:
8737:
8728:
8725:
8700:
8696:
8692:
8687:
8683:
8671:
8668:
8660:
8645:
8642:
8639:
8631:
8612:
8604:
8599:
8593:
8590:
8584:
8579:
8574:
8570:
8562:
8559:
8539:
8536:
8533:
8527:
8518:
8514:
8504:
8501:
8495:
8488:
8485:
8480:
8475:
8471:
8441:
8433:
8432:
8425:
8422:
8414:
8404:
8400:
8394:
8393:
8388:This section
8386:
8382:
8377:
8376:
8368:
8345:
8330:
8327:
8323:
8318:
8315:
8305:
8297:
8293:
8292:
8290:
8274:
8268:
8265:
8261:
8252:and variance
8236:
8233:
8229:
8224:
8221:
8198:
8193:
8189:
8185:
8159:
8151:
8147:
8143:
8140:
8132:
8129:
8111:
8108:
8105:
8102:
8078:
8075:
8050:
8042:
8038:
8034:
8031:
8023:
8022:
8021:
8018:
8016:
7998:
7994:
7968:
7964:
7957:
7954:
7945:
7931:
7927:
7923:
7915:
7897:
7893:
7889:
7879:
7875:
7858:
7855:
7849:
7842:
7839:
7836:
7813:
7790:
7782:
7778:
7774:
7771:
7763:
7747:
7744:
7741:
7721:
7701:
7693:
7676:
7667:
7650:
7646:
7642:
7639:
7636:
7613:
7589:
7585:
7581:
7577:
7573:
7570:
7567:
7556:
7552:
7548:
7545:
7540:
7537:
7534:
7530:
7526:
7522:
7518:
7515:
7512:
7509:
7504:
7501:
7498:
7494:
7490:
7486:
7482:
7479:
7476:
7470:
7465:
7457:
7448:
7445:
7423:
7420:
7417:
7413:
7409:
7405:
7397:with a width
7384:
7364:
7342:
7313:
7305:
7302:
7299:
7293:
7290:
7287:
7261:
7258:
7252:
7249:
7246:
7238:
7235:
7230:
7227:
7224:
7221:
7218:
7212:
7202:
7185:
7182:
7176:
7173:
7170:
7164:
7161:
7158:
7153:
7150:
7145:
7142:
7139:
7136:
7133:
7127:
7117:
7116:
7115:
7110:Concentration
7093:
7089:
7083:
7080:
7077:
7069:
7066:
7063:
7059:
7055:
7050:
7046:
7038:
7037:
7036:
7034:
7030:
7006:
6999:
6994:
6991:
6986:
6976:
6970:
6967:
6962:
6959:
6955:
6943:
6939:
6935:
6929:
6926:
6923:
6920:
6917:
6914:
6908:
6902:
6899:
6896:
6887:
6884:
6881:
6875:
6872:
6864:
6860:
6853:
6843:
6842:
6841:
6827:
6813:
6811:
6807:
6788:
6782:
6779:
6776:
6770:
6766:
6762:
6756:
6753:
6744:
6738:
6735:
6729:
6706:
6703:
6697:
6691:
6668:
6660:
6655:
6653:
6634:
6628:
6605:
6601:
6596:
6593:
6588:
6583:
6578:
6572:
6569:
6564:
6561:
6557:
6553:
6549:
6544:
6539:
6536:
6531:
6523:
6519:
6515:
6512:
6509:
6504:
6501:
6496:
6493:
6490:
6483:
6479:
6476:
6470:
6467:
6464:
6458:
6454:
6451:
6445:
6435:
6431:
6427:
6424:
6417:
6416:
6415:
6413:
6403:
6387:
6383:
6380:
6374:
6369:
6365:
6339:
6317:
6313:
6293:
6273:
6265:
6260:
6244:
6240:
6236:
6232:
6229:
6224:
6220:
6216:
6213:
6210:
6207:
6201:
6190:
6186:
6177:
6167:
6151:
6147:
6142:
6139:
6134:
6130:
6122:and variance
6109:
6106:
6103:
6100:
6097:
6077:
6069:
6048:
6040:
6036:
6032:
6027:
6023:
6012:
6008:
6004:
6000:
5997:
5994:
5991:
5988:
5984:
5980:
5976:
5973:
5970:
5966:
5962:
5959:
5956:
5951:
5947:
5941:
5936:
5933:
5930:
5926:
5920:
5917:
5912:
5904:
5895:
5894:
5893:
5879:
5859:
5839:
5831:
5816:
5802:
5786:
5782:
5778:
5775:
5772:
5767:
5763:
5740:
5736:
5732:
5729:
5726:
5721:
5717:
5713:
5710:
5684:
5681:
5678:
5672:
5669:
5647:
5643:
5616:
5613:
5610:
5604:
5601:
5598:
5593:
5589:
5572:
5558:
5553:
5548:
5545:
5542:
5538:
5531:
5523:
5518:
5514:
5510:
5507:
5504:
5501:
5498:
5493:
5488:
5484:
5477:
5471:
5468:
5465:
5455:
5444:
5437:
5431:
5428:
5418:
5411:
5403:
5389:
5383:
5378:
5374:
5365:
5360:
5357:
5354:
5350:
5326:
5321:
5318:
5309:
5293:
5287:
5284:
5279:
5269:
5265:
5256:
5240:
5237:
5227:
5223:
5220:
5195:
5191:
5187:
5184:
5181:
5178:
5175:
5170:
5166:
5159:
5156:
5136:
5114:
5110:
5087:
5083:
5079:
5076:
5073:
5070:
5067:
5062:
5058:
5050:eigenvectors
5037:
5034:
5031:
5011:
4985:
4974:
4970:
4963:
4958:
4954:
4933:
4907:
4884:
4879:
4859:
4856:
4846:
4839:
4833:
4815:
4802:
4792:
4789:
4782:
4779:
4774:
4762:
4755:
4747:
4737:
4730:
4727:
4722:
4717:
4713:
4707:
4702:
4699:
4696:
4692:
4685:
4677:
4663:
4657:
4652:
4648:
4639:
4634:
4631:
4628:
4624:
4597:
4574:
4551:
4546:
4543:
4534:
4518:
4515:
4507:
4501:
4488:
4483:
4479:
4473:
4468:
4465:
4462:
4458:
4452:
4449:
4444:
4435:
4410:
4405:
4402:
4399:
4395:
4391:
4386:
4372:
4366:
4361:
4357:
4348:
4343:
4340:
4337:
4333:
4324:
4320:
4302:
4298:
4294:
4291:
4288:
4285:
4282:
4277:
4273:
4264:
4260:
4256:
4241:
4239:
4235:
4234:approximation
4216:
4211:
4207:
4203:
4193:
4190:
4187:
4183:
4179:
4173:
4167:
4163:
4160:
4157:
4151:
4148:
4145:
4138:
4137:
4136:
4122:
4119:
4116:
4107:
4094:
4089:
4085:
4081:
4071:
4068:
4065:
4061:
4057:
4051:
4045:
4041:
4038:
4035:
4029:
4009:
4006:
4003:
4000:
3997:
3989:
3973:
3969:
3965:
3962:
3959:
3950:
3948:
3944:
3939:
3925:
3902:
3898:
3895:
3889:
3867:
3863:
3859:
3856:
3852:
3846:
3843:
3838:
3832:
3828:
3825:
3819:
3794:
3790:
3786:
3783:
3779:
3775:
3772:
3769:
3763:
3759:
3756:
3750:
3743:
3742:
3741:
3727:
3724:
3721:
3712:
3710:
3691:
3688:
3685:
3679:
3671:
3652:
3649:
3646:
3640:
3617:
3613:
3607:
3604:
3598:
3593:
3590:
3584:
3580:
3577:
3566:
3563:
3545:
3542:
3536:
3531:
3528:
3520:
3514:
3508:
3504:
3501:
3495:
3488:
3487:
3486:
3484:
3464:
3461:
3458:
3450:
3445:
3436:
3434:
3418:
3409:
3407:
3394:
3384:
3365:
3361:
3357:
3321:
3311:
3308:
3301:
3298:
3295:
3292:
3287:
3279:
3274:
3271:
3266:
3257:
3253:
3249:
3245:
3237:
3233:
3229:
3226:
3222:
3216:
3213:
3210:
3206:
3202:
3198:
3187:
3182:
3176:
3172:
3169:
3163:
3156:
3155:
3154:
3152:
3142:
3128:
3119:
3105:
3083:
3079:
3058:
3038:
3016:
3012:
3008:
3005:
3000:
2996:
2975:
2953:
2949:
2926:
2922:
2899:
2895:
2886:
2865:
2861:
2854:
2844:
2840:
2836:
2831:
2827:
2815:
2810:
2807:
2804:
2800:
2796:
2791:
2787:
2779:
2778:
2777:
2775:
2753:
2750:
2743:
2735:
2732:
2729:
2726:
2723:
2720:
2711:
2705:
2702:
2695:
2687:
2684:
2681:
2678:
2669:
2664:
2660:
2652:
2651:
2650:
2636:
2633:
2630:
2627:
2624:
2601:
2598:
2595:
2589:
2586:
2583:
2580:
2557:
2554:
2551:
2545:
2542:
2539:
2536:
2533:
2530:
2504:
2501:
2498:
2491:
2483:
2480:
2477:
2474:
2465:
2460:
2456:
2448:
2447:
2446:
2443:
2429:
2426:
2423:
2420:
2417:
2397:
2377:
2357:
2330:
2327:
2324:
2318:
2315:
2312:
2309:
2303:
2300:
2293:
2292:
2291:
2287:
2285:
2284:binomial test
2281:
2276:
2272:
2268:
2263:
2247:
2242:
2238:
2231:
2225:
2202:
2180:
2176:
2172:
2169:
2146:
2143:
2140:
2134:
2131:
2128:
2108:
2088:
2068:
2061:Suppose that
2059:
2053:
2049:
2041:
2037:
2035:
2028:
2026:
2022:
2021:-distribution
2020:
2015:
2014:-distribution
2013:
2005:
2002:
2000:
1997:
1994:
1991:
1988:
1987:Log-rank test
1985:
1982:
1979:
1976:
1973:
1971:
1967:
1964:
1963:
1962:
1960:
1956:
1945:
1942:
1938:
1934:
1911:
1906:
1901:
1897:
1890:
1884:
1861:
1853:
1849:
1842:
1836:
1829:
1828:
1827:
1806:
1801:
1796:
1792:
1786:
1781:
1778:
1775:
1771:
1767:
1761:
1754:
1753:
1752:
1750:
1746:
1740:
1736:
1729:
1718:
1716:
1712:
1708:
1704:
1700:
1696:
1691:
1689:
1685:
1681:
1677:
1674:, notably in
1673:
1669:
1664:
1647:
1644:
1639:
1635:
1626:
1616:
1613:
1588:
1584:
1580:
1577:
1574:
1571:
1566:
1563:
1558:
1555:
1544:
1541:
1519:
1514:
1510:
1504:
1500:
1496:
1493:
1485:
1481:
1463:
1458:
1454:
1448:
1444:
1436:
1431:
1414:
1411:
1408:
1400:
1390:
1387:
1367:
1347:
1324:
1321:
1318:
1315:
1310:
1307:
1302:
1299:
1288:
1285:
1263:
1258:
1254:
1250:
1247:
1239:
1235:
1217:
1212:
1208:
1198:
1196:
1193:
1178:
1170:
1155:
1147:
1146:-distribution
1131:
1127:
1117:
1113:
1109:
1105:
1081:
1076:
1073:
1070:
1067:
1057:
1053:
1049:
1046:
1038:
1035:
1032:
1029:
1026:
1023:
1011:
1007:
990:
986:
982:
979:
971:
968:
965:
962:
959:
947:
943:
925:
922:
917:
914:
904:
900:
896:
893:
885:
882:
879:
876:
864:
860:
840:
835:
832:
827:
823:
819:
813:
810:
805:
802:
798:
794:
791:
782:
769:
766:
751:
747:
743:
740:
737:
735:
728:
725:
710:
706:
689:
686:
675:
671:
653:
649:
645:
634:
630:
614:
611:
602:
598:
579:
576:
573:
570:
567:
552:
548:
530:
515:
512:
508:
503:
500:
490:
487:
478:
474:
459:
450:
446:
429:
423:
420:
414:
409:
406:
400:
396:
386:
382:
378:
368:
357:
353:
335:
331:
327:
324:
320:
314:
311:
308:
304:
300:
296:
285:
281:
277:
266:
262:
258:
254:
249:
238:
234:
212:
209:
206:
200:
197:
188:
184:
160:
150:
147:
138:
134:
116:
111:
107:
82:
74:
70:
58:
54:
46:
42:
34:
26:
22:
16788:
16776:
16742:Multivariate
16741:
16729:
16717:
16712:Wrapped Lévy
16672:
16620:Matrix gamma
16613:
16593:
16581:Normal-gamma
16574:
16540:Continuous:
16539:
16510:
16455:Tukey lambda
16442:
16434:
16429:-exponential
16426:
16418:
16409:
16400:
16391:
16385:-exponential
16382:
16326:
16293:
16260:
16222:
16209:
16136:Poly-Weibull
16081:Log-logistic
16041:
16040:Hotelling's
15972:
15903:
15814:Logit-normal
15688:Gauss–Kuzmin
15683:Flory–Schulz
15564:with finite
15494:
15490:
15455:
15451:
15435:
15415:
15411:
15385:
15382:Hald, Anders
15339:
15335:
15325:
15309:
15304:
15293:
15281:
15257:
15248:
15228:
15218:
15213:
15170:
15166:
15160:
15138:(1): 19–30.
15135:
15131:
15121:
15076:
15070:
15060:
15025:
15019:
14985:
14981:
14975:
14950:
14942:
14907:
14903:
14893:
14878:
14873:
14857:
14846:. Retrieved
14829:(1): 60–65.
14826:
14822:
14809:
14800:
14766:
14762:
14756:
14737:
14731:
14712:
14706:
14687:
14664:
14620:
14616:"Chapter 26"
14602:
14591:. Retrieved
14584:the original
14570:
14553:denotes the
14273:is given as
14164:
14160:
14143:Pearson 1914
14127:Karl Pearson
14124:
14120:Helmert'sche
14119:
14113:
14101:
14095:
14079:
14068:
14027:
13577:
13575:
13566:
13562:
13559:less extreme
13558:
13554:
13546:
13521:
13513:
13512:
13506:
13490:
12703:
12668:
12522:
12427:Because the
12426:
12266:
12246:
12230:
12228:
12027:
12014:
11451:
10995:
10805:
10796:
10790:
10594:
10588:independent
10547:
10360:follows the
9062:
9058:
8417:
8408:
8397:Please help
8392:verification
8389:
8366:
8288:
8128:R. A. Fisher
8019:
7946:
7689:
7279:
7113:
7033:power series
7026:
6819:
6656:
6620:
6414:is given by
6409:
6261:
6168:
6066:
5892:parameters:
5808:
5580:
4505:
4504:
4262:
4261:
4258:
4232:For another
4231:
4108:
3951:
3943:spreadsheets
3940:
3811:
3713:
3632:
3480:
3410:
3385:, which has
3381:denotes the
3343:
3148:
3120:
2887:
2884:
2774:Karl Pearson
2771:
2522:
2444:
2349:
2288:
2274:
2264:
2060:
2039:
2033:
2029:
2024:
2018:
2011:
2008:
1951:
1948:Introduction
1940:
1936:
1926:
1821:
1738:
1734:
1727:
1724:
1703:independence
1692:
1683:
1665:
1434:
1432:
1199:
1119:
1115:
1111:
1101:
16826:Exponential
16675:directional
16664:Directional
16551:Generalized
16522:Multinomial
16477:continuous-
16417:Kaniadakis
16408:Kaniadakis
16399:Kaniadakis
16390:Kaniadakis
16381:Kaniadakis
16333:Tracy–Widom
16310:Skew normal
16292:Noncentral
16076:Log-Laplace
16054:Generalized
16035:Half-normal
16001:Generalized
15965:Logarithmic
15950:Exponential
15904:Chi-squared
15844:U-quadratic
15809:Kumaraswamy
15751:Continuous
15698:Logarithmic
15593:Categorical
15491:Mathematica
10931:, then the
6019:where
5805:Sample mean
5342:, we have
4319:independent
1745:independent
1721:Definitions
1192:independent
31:Chi-squared
25:Chi2 (band)
16904:Categories
16821:Elliptical
16777:Degenerate
16763:Degenerate
16511:Discrete:
16470:univariate
16325:Student's
16280:Asymmetric
16259:Johnson's
16187:supported
16131:Phase-type
16086:Log-normal
16071:Log-Cauchy
16061:Kolmogorov
15979:Noncentral
15909:Noncentral
15889:Beta prime
15839:Triangular
15834:Reciprocal
15804:Irwin–Hall
15753:univariate
15733:Yule–Simon
15615:Rademacher
15557:univariate
15458:: 85–154.
15452:Biometrika
15412:Biometrika
15035:1505.01957
14848:2012-05-01
14593:2009-03-06
14563:References
14153:, writing
13510:values vs
13109:Statistic
13006:statistics
12683:regression
12671:statistics
12565:with even
12309:, in that
11143:such that
9386:for more.)
7806:, then as
6681:for which
5872:and scale
5577:Additivity
4244:Properties
2036:-statistic
2004:Score test
1116:chi-square
1108:statistics
137:Parameters
16546:Dirichlet
16527:Dirichlet
16437:-Gaussian
16412:-Logistic
16249:Holtsmark
16221:Gaussian
16208:Fisher's
16191:real line
15693:Geometric
15673:Delaporte
15578:Bernoulli
15555:Discrete
15442:EMS Press
15364:237919587
15356:0361-0926
15274:Hald 1998
15205:119721108
15180:1208.2691
14934:116945590
14926:0090-5364
14866:MathWorld
14489:α
14466:Ψ
14441:α
14435:Ψ
14403:β
14399:γ
14386:α
14375:Ψ
14364:γ
14348:β
14345:−
14339:
14328:−
14325:α
14307:α
14303:β
14258:∞
14077:ICDF for
13624:χ
13589:χ
13504:Table of
13454:σ
13416:∑
13365:σ
13354:μ
13350:−
13312:∑
13263:σ
13225:∑
13178:σ
13167:μ
13163:−
13125:∑
13081:…
13049:σ
13036:μ
13026:∼
13008:based on
12962:∑
12943:¯
12901:−
12894:χ
12884:σ
12880:∼
12865:¯
12850:−
12817:∑
12784:σ
12777:μ
12679:variances
12542:χ
12538:∼
12486:
12480:∼
12446:χ
12442:∼
12382:Γ
12379:∼
12323:Γ
12320:∼
12282:χ
12278:∼
12153:∑
12111:∈
12095:…
12049:…
11869:χ
11865:∼
11804:χ
11800:∼
11752:χ
11748:∼
11697:χ
11693:∼
11645:χ
11641:∼
11490:∼
11423:χ
11419:∼
11411:⊤
11375:…
11339:Σ
11304:…
11234:…
11213:≥
11164:⋯
11103:a random
11068:Σ
11053:∼
11027:×
11004:Σ
10978:−
10916:−
10896:symmetric
10880:×
10753:μ
10750:−
10736:−
10718:μ
10715:−
10655:and rank
10643:μ
10489:β
10474:χ
10470:∼
10465:α
10459:β
10449:μ
10446:−
10405:∑
10384:β
10378:α
10372:μ
10304:χ
10300:∼
10295:β
10286:μ
10283:−
10242:∑
10217:β
10211:μ
10205:
10199:∼
10155:χ
10151:∼
10139:
10130:−
10090:
10084:∼
10038:ν
10016:ν
10006:
10000:∼
9946:ν
9941:χ
9937:∼
9900:ν
9895:χ
9891:∼
9847:ν
9843:χ
9839:
9822:∼
9783:ν
9779:χ
9775:∼
9738:χ
9734:∼
9690:
9684:∼
9647:χ
9643:∼
9599:
9593:∼
9554:χ
9550:∼
9544:λ
9518:λ
9506:
9500:∼
9447:
9441:∼
9404:χ
9400:∼
9349:
9343:∼
9309:χ
9305:∼
9269:χ
9265:∼
9227:χ
9223:∼
9180:θ
9166:ν
9154:Γ
9151:∼
9090:ν
9086:χ
9082:∼
9030:χ
9026:∼
9017:‖
8990:…
8968:‖
8932:χ
8905:∞
8902:→
8893:ν
8851:ν
8833:∼
8791:ν
8786:χ
8756:ν
8750:∞
8747:→
8738:ν
8697:ν
8684:ν
8672:∼
8640:λ
8591:χ
8585:∼
8571:χ
8486:−
8472:χ
8448:∞
8445:→
8319:−
8225:−
8148:χ
8144:∼
8109:−
8039:χ
8035:∼
8013:, as the
7995:χ
7965:χ
7958:
7840:−
7779:χ
7775:∼
7614:α
7590:α
7582:−
7574:−
7568:≥
7557:α
7541:α
7505:α
7480:−
7471:∈
7462:‖
7455:‖
7424:α
7291:∼
7259:−
7253:
7247:≤
7228:≥
7222:−
7213:
7183:−
7177:
7171:≤
7143:≥
7137:−
7128:
7081:−
7067:−
7047:κ
7029:cumulants
7023:Cumulants
6983:Γ
6952:Γ
6927:−
6909:⋯
6854:
6783:
6757:ψ
6739:
6730:
6692:
6629:ψ
6584:ψ
6565:−
6528:Γ
6516:
6468:
6441:∞
6432:∫
6366:σ
6343:¯
6221:σ
6208:μ
6197:∞
6194:→
6181:¯
6148:θ
6143:α
6131:σ
6110:θ
6107:⋅
6104:α
6098:μ
6078:α
6037:χ
6033:∼
5995:θ
5971:α
5963:
5957:∼
5927:∑
5908:¯
5880:θ
5860:α
5776:⋯
5730:⋯
5689:¯
5621:¯
5546:−
5539:χ
5532:∼
5460:⊤
5449:⊤
5423:⊤
5393:¯
5384:−
5351:∑
5313:¯
5274:⊤
5260:¯
5241:∼
5232:⊤
5035:−
4989:¯
4911:¯
4851:⊤
4826:⊤
4819:¯
4806:¯
4783:−
4767:⊤
4741:¯
4728:−
4693:∑
4667:¯
4658:−
4625:∑
4601:¯
4538:¯
4519:∼
4459:∑
4439:¯
4403:−
4396:χ
4392:∼
4376:¯
4367:−
4334:∑
4191:−
4174:≤
4149:−
4069:−
4052:≤
3963:≡
3857:−
3784:−
3776:−
3641:γ
3556:Γ
3521:γ
3352:Γ
3318:otherwise
3263:Γ
3227:−
3214:−
2923:χ
2896:χ
2837:−
2801:∑
2788:χ
2730:−
2724:−
2682:−
2661:χ
2634:−
2599:−
2555:−
2478:−
2457:χ
2427:−
2313:−
2301:χ
2239:χ
2232:∼
2132:∼
1999:Wald test
1898:χ
1891:∼
1850:χ
1843:∼
1772:∑
1617:∼
1575:θ
1556:α
1545:∼
1511:χ
1497:∼
1455:χ
1391:∼
1368:θ
1348:α
1319:θ
1300:α
1289:∼
1255:χ
1251:∼
1209:χ
1128:χ
1047:−
1036:
1027:−
980:−
963:−
894:−
880:−
824:ψ
806:−
755:Γ
744:
571:−
504:−
488:≈
397:γ
373:Γ
325:−
312:−
272:Γ
216:∞
201:∈
161:∗
151:∈
108:χ
71:χ
16879:Category
16811:Circular
16804:Families
16789:Singular
16768:singular
16532:Negative
16479:discrete
16445:-Weibull
16403:-Weibull
16287:Logistic
16171:Discrete
16141:Rayleigh
16121:Nakagami
16044:-squared
16018:Gompertz
15867:interval
15603:Negative
15588:Binomial
15406:(1902).
15384:(1998).
15236:Archived
15152:19777585
15113:16577411
15052:31582370
14843:10327785
14656:65-12253
14640:64-60036
14427:, where
14176:See also
13547:at least
11623:, where
9596:Rayleigh
8594:′
8515:→
7912:and the
7878:skewness
6187:→
4263:Theorem.
3952:Letting
3945:and all
2023:used in
2016:and the
1957:and the
633:Skewness
601:Variance
60:Notation
16889:Commons
16861:Wrapped
16856:Tweedie
16851:Pearson
16846:Mixture
16753:Bingham
16652:Complex
16642:Inverse
16632:Wishart
16625:Inverse
16612:Matrix
16586:Inverse
16502:(joint)
16421:-Erlang
16275:Laplace
16166:Weibull
16023:Shifted
16006:Inverse
15991:Fréchet
15914:Inverse
15849:Uniform
15769:Arcsine
15728:Skellam
15723:Poisson
15646:support
15620:Soliton
15573:Benford
15566:support
15444:, 2001
15185:Bibcode
15104:1076144
15081:Bibcode
15002:2983618
14864:, from
14803:, Wiley
14783:1164752
14648:0167642
14110:History
14099:is the
14088:yields
13518:-values
12808:, then
12585:, then
12469:, then
12233:means.
12231:nonzero
10695:, then
10202:Laplace
10122:) then
9716:) then
9687:Maxwell
9625:) then
9533:, then
9430:, then
9382:. (See
9332:, then
9137:, then
7690:By the
6650:is the
6406:Entropy
5024:, and
3707:is the
3668:is the
1733:, ...,
1340:(where
1148:) with
709:Entropy
187:Support
16795:Cantor
16637:Normal
16468:Mixed
16394:-Gamma
16320:Stable
16270:Landau
16244:Gumbel
16198:Cauchy
16126:Pareto
15938:Erlang
15919:Scaled
15874:Benini
15713:Panjer
15392:
15362:
15354:
15203:
15150:
15111:
15101:
15050:
15000:
14963:
14932:
14924:
14885:
14841:
14781:
14744:
14719:
14694:
14654:
14646:
14638:
14628:
14104:-value
14086:df = 7
14082:= 0.05
14063:0.001
14022:29.59
14019:23.21
14016:18.31
14013:15.99
14010:13.44
14007:11.78
13984:27.88
13981:21.67
13978:16.92
13975:14.68
13972:12.24
13969:10.66
13946:26.12
13943:20.09
13940:15.51
13937:13.36
13934:11.03
13908:24.32
13905:18.48
13902:14.07
13899:12.02
13870:22.46
13867:16.81
13864:12.59
13861:10.64
13832:20.52
13829:15.09
13826:11.07
13794:18.47
13791:13.28
13756:16.27
13753:11.34
13718:13.82
13680:10.83
13650:0.004
13642:value
13543:-value
12923:where
12762:i.i.d.
12516:is an
12251:where
10847:i.i.d.
9503:Erlang
9482:is an
9444:Erlang
9378:is an
8525:
8510:
6621:where
6357:being
5830:i.i.d.
5535:
5529:
5481:
5475:
5441:
5435:
5415:
5409:
4872:where
4843:
4837:
4759:
4753:
4689:
4683:
4506:Proof.
4425:where
3633:where
3344:where
3098:; and
2885:where
2617:, and
2523:Using
2410:, and
2350:where
2235:
2229:
2223:
1894:
1888:
1882:
1879:
1871:
1868:
1846:
1840:
1765:
1114:(also
1110:, the
477:Median
169:
166:
16517:Ewens
16343:Voigt
16315:Slash
16096:Lomax
16091:Log-t
15996:Gamma
15943:Hyper
15933:Davis
15928:Dagum
15784:Bates
15774:ARGUS
15658:Borel
15360:S2CID
15201:S2CID
15175:arXiv
15148:S2CID
15048:S2CID
15030:arXiv
14998:JSTOR
14930:S2CID
14910:(5).
14839:S2CID
14819:(PDF)
14779:JSTOR
14587:(PDF)
14580:(PDF)
14060:0.01
14057:0.05
14054:0.10
14051:0.20
14048:0.30
14045:0.50
14042:0.70
14039:0.80
14036:0.90
14033:0.95
14004:9.34
14001:7.27
13998:6.18
13995:4.87
13992:3.94
13966:8.34
13963:6.39
13960:5.38
13957:4.17
13954:3.32
13931:9.52
13928:7.34
13925:5.53
13922:4.59
13919:3.49
13916:2.73
13896:9.80
13893:8.38
13890:6.35
13887:4.67
13884:3.82
13881:2.83
13878:2.17
13858:8.56
13855:7.23
13852:5.35
13849:3.83
13846:3.07
13843:2.20
13840:1.63
13823:9.24
13820:7.29
13817:6.06
13814:4.35
13811:3.00
13808:2.34
13805:1.61
13802:1.14
13788:9.49
13785:7.78
13782:5.99
13779:4.88
13776:3.36
13773:2.20
13770:1.65
13767:1.06
13764:0.71
13750:7.81
13747:6.25
13744:4.64
13741:3.66
13738:2.37
13735:1.42
13732:1.01
13729:0.58
13726:0.35
13715:9.21
13712:5.99
13709:4.61
13706:3.22
13703:2.41
13700:1.39
13697:0.71
13694:0.45
13691:0.21
13688:0.10
13677:6.63
13674:3.84
13671:2.71
13668:1.64
13665:1.07
13662:0.46
13659:0.15
13656:0.06
13653:0.02
11912:. If
11255:then
11016:is a
10902:with
10869:is a
10615:is a
10396:then
10233:then
9802:then
9250:then
8873:then
8718:then
8294:This
8175:then
8066:then
7874:tends
5960:Gamma
2038:in a
1549:Gamma
1534:then
1293:Gamma
1278:then
16766:and
16724:Kent
16151:Rice
16066:Lévy
15894:Burr
15824:PERT
15789:Beta
15738:Zeta
15630:Zipf
15547:list
15390:ISBN
15352:ISSN
15314:61f.
15243:4.61
15109:PMID
14961:ISBN
14922:ISSN
14883:ISBN
14742:ISBN
14717:ISBN
14692:ISBN
14652:LCCN
14636:LCCN
14626:ISBN
14094:1 –
14084:and
13555:(df)
13522:The
13106:Name
12760:are
12523:The
12249:z'Az
12121:>
11939:and
11782:and
11675:and
11195:and
11083:and
10904:rank
10003:Beta
9926:and
9121:>
9110:and
9055:norm
7745:>
7680:blue
7027:The
6719:and
6410:The
4508:Let
4317:are
4120:>
4007:<
4001:<
3672:and
3485:is:
3481:Its
3296:>
3149:The
1944:s).
1743:are
1697:for
1606:and
1433:The
1106:and
1077:<
1071:<
918:<
551:Mode
449:Mean
16602:LKJ
15899:Chi
15460:doi
15420:doi
15344:doi
15292:",
15288:, "
15193:doi
15140:doi
15099:PMC
15089:doi
15040:doi
14990:doi
14957:118
14912:doi
14831:doi
14771:doi
14336:exp
14155:−½χ
14151:Chi
13989:10
12708:if
12483:Exp
12028:If
11730:If
11534:if
11475:is
10996:If
10806:If
10797:not
10595:If
10333:If
10181:If
10136:log
10073:If
9880:If
9764:If
9673:If
9582:If
9489:If
9389:If
9346:Exp
9294:If
9212:If
9205:. (
9071:If
9057:of
8888:lim
8733:lim
8661:If
8434:As
8401:by
8133:If
8024:If
7916:is
7880:is
7684:red
7250:exp
7174:exp
6402:).
4265:If
3449:CDF
1725:If
1670:in
1118:or
1102:In
1010:PGF
863:MGF
741:log
562:max
356:CDF
237:PDF
99:or
16906::
15456:10
15454:.
15440:,
15434:,
15414:.
15410:.
15358:.
15350:.
15340:52
15338:.
15334:.
15297:21
15266:^
15256:.
15221:,
15199:.
15191:.
15183:.
15171:46
15169:.
15146:.
15136:26
15134:.
15130:.
15107:.
15097:.
15087:.
15077:17
15075:.
15069:.
15046:.
15038:.
15026:44
15024:.
15010:^
14996:.
14984:.
14959:.
14928:.
14920:.
14908:28
14906:.
14902:.
14837:.
14827:22
14825:.
14821:.
14791:^
14777:.
14767:13
14765:.
14676:^
14650:.
14644:MR
14642:.
14634:.
14618:.
14610:;
14159:−½
13951:9
13913:8
13875:7
13837:6
13799:5
13761:4
13723:3
13685:2
13647:1
13495:.
12661:.
12520:.
11479:,
10898:,
10803:.
8460:,
7955:ln
7944:.
7924:12
7748:50
7666:.
6812:.
6780:ln
6736:ln
6654:.
6513:ln
6465:ln
5662:,
5224::=
5160::=
4964::=
4840:=:
4240:.
3986:,
3949:.
3938:.
3711:.
3465:10
3435:.
3408:.
2988:;
2573:,
2442:.
2121::
1875:or
1747:,
1717:.
1690:.
1663:.
1430:.
1033:ln
946:CF
687:12
16614:t
16575:t
16443:q
16435:q
16427:q
16419:κ
16410:κ
16401:κ
16392:κ
16383:κ
16327:t
16294:t
16263:U
16261:S
16223:q
16210:z
16042:T
15973:F
15549:)
15545:(
15535:e
15528:t
15521:v
15495:x
15466:.
15462::
15426:.
15422::
15416:1
15398:.
15366:.
15346::
15320:.
15260:.
15207:.
15195::
15187::
15177::
15154:.
15142::
15115:.
15091::
15083::
15054:.
15042::
15032::
15004:.
14992::
14986:8
14969:.
14936:.
14914::
14851:.
14833::
14785:.
14773::
14750:.
14725:.
14700:.
14658:.
14596:.
14557:.
14540:)
14536:z
14533:;
14526:)
14523:0
14520:,
14517:1
14514:(
14506:)
14500:2
14497:1
14492:,
14485:(
14476:(
14470:1
14460:1
14453:=
14450:)
14447:z
14444:,
14438:(
14410:)
14394:,
14389:2
14380:(
14370:)
14367:x
14361:+
14356:2
14352:x
14342:(
14331:1
14321:x
14315:2
14311:/
14299:2
14293:=
14290:)
14287:x
14284:(
14281:f
14261:)
14255:,
14252:0
14249:(
14165:x
14163:Σ
14161:x
14102:p
14096:p
14080:p
14075:χ
14028:p
13628:2
13593:2
13578:p
13567:p
13563:p
13531:p
13514:p
13507:χ
13470:2
13465:)
13458:i
13448:i
13444:X
13438:(
13431:k
13426:1
13423:=
13420:i
13381:2
13376:)
13369:i
13358:i
13345:i
13341:X
13334:(
13327:k
13322:1
13319:=
13316:i
13279:2
13274:)
13267:i
13257:i
13253:X
13247:(
13240:k
13235:1
13232:=
13229:i
13194:2
13189:)
13182:i
13171:i
13158:i
13154:X
13147:(
13140:k
13135:1
13132:=
13129:i
13087:k
13084:,
13078:,
13075:1
13072:=
13069:i
13066:,
13063:)
13058:2
13053:i
13045:,
13040:i
13032:(
13029:N
13021:i
13017:X
13001:.
12987:i
12983:X
12977:n
12972:1
12969:=
12966:i
12956:n
12953:1
12948:=
12938:i
12934:X
12909:2
12904:1
12898:n
12888:2
12875:2
12871:)
12860:i
12856:X
12845:i
12841:X
12837:(
12832:n
12827:1
12824:=
12821:i
12793:)
12788:2
12780:,
12774:(
12771:N
12746:n
12742:X
12738:,
12735:.
12732:.
12729:.
12726:,
12721:1
12717:X
12649:2
12645:/
12641:1
12621:2
12617:/
12613:k
12593:X
12573:k
12551:2
12546:k
12535:X
12503:)
12498:2
12495:1
12490:(
12477:X
12455:2
12450:2
12439:X
12422:k
12407:)
12403:2
12400:,
12395:2
12392:k
12386:(
12376:X
12355:)
12349:2
12346:1
12341:,
12336:2
12333:k
12327:(
12317:X
12291:2
12286:k
12275:X
12257:A
12253:z
12188:i
12184:X
12178:i
12174:a
12168:n
12163:1
12160:=
12157:i
12149:=
12146:X
12124:0
12116:R
12106:n
12102:a
12098:,
12092:,
12087:1
12083:a
12060:n
12056:X
12052:,
12046:,
12041:1
12037:X
12017:k
11992:2
11988:X
11984:+
11979:1
11975:X
11952:2
11948:X
11925:1
11921:X
11898:2
11891:2
11887:k
11883:+
11878:1
11874:k
11860:2
11856:X
11852:+
11847:1
11843:X
11820:2
11813:2
11809:k
11795:2
11791:X
11768:2
11761:1
11757:k
11743:1
11739:X
11713:2
11706:2
11702:k
11688:2
11684:X
11661:2
11654:1
11650:k
11636:1
11632:X
11605:2
11601:k
11595:/
11588:2
11584:X
11574:1
11570:k
11564:/
11557:1
11553:X
11545:=
11542:Y
11522:)
11517:2
11513:k
11509:,
11504:1
11500:k
11496:(
11493:F
11487:Y
11463:Y
11437:.
11432:2
11427:1
11406:)
11398:p
11394:X
11388:p
11384:w
11378:,
11372:,
11365:1
11361:X
11355:1
11351:w
11344:(
11335:)
11327:p
11323:X
11317:p
11313:w
11307:,
11301:,
11294:1
11290:X
11284:1
11280:w
11273:(
11268:1
11243:,
11240:p
11237:,
11231:,
11228:1
11225:=
11222:i
11219:,
11216:0
11208:i
11204:w
11183:1
11180:=
11175:p
11171:w
11167:+
11161:+
11156:1
11152:w
11131:X
11111:p
11091:w
11071:)
11065:,
11062:0
11059:(
11056:N
11050:X
11030:p
11024:p
10981:n
10975:k
10955:Y
10952:A
10947:T
10943:Y
10919:n
10913:k
10883:k
10877:k
10857:A
10834:k
10814:Y
10776:k
10756:)
10747:Y
10744:(
10739:1
10732:C
10726:T
10722:)
10712:Y
10709:(
10706:=
10703:X
10683:C
10663:k
10623:k
10603:Y
10576:k
10556:k
10494:2
10485:/
10481:n
10478:2
10454:|
10441:i
10437:X
10432:|
10428:2
10420:n
10415:1
10412:=
10409:i
10381:,
10375:,
10346:i
10342:X
10316:2
10311:n
10308:2
10290:|
10278:i
10274:X
10269:|
10265:2
10257:n
10252:1
10249:=
10246:i
10220:)
10214:,
10208:(
10194:i
10190:X
10164:2
10159:2
10148:)
10145:X
10142:(
10133:2
10118:(
10105:)
10102:1
10099:,
10096:0
10093:(
10087:U
10081:X
10070:)
10066:(
10053:)
10047:2
10042:2
10031:,
10025:2
10020:1
10009:(
9993:Y
9990:+
9987:X
9983:X
9957:2
9950:2
9934:Y
9911:2
9904:1
9888:X
9871:)
9867:(
9852:2
9835:-
9832:v
9829:n
9826:I
9816:X
9813:1
9788:2
9772:X
9747:2
9742:3
9729:2
9725:X
9712:(
9699:)
9696:1
9693:(
9681:X
9656:2
9651:2
9638:2
9634:X
9621:(
9608:)
9605:1
9602:(
9590:X
9566:2
9561:k
9558:2
9547:X
9541:2
9521:)
9515:,
9512:k
9509:(
9497:X
9486:.
9470:)
9467:2
9463:/
9459:1
9456:,
9453:k
9450:(
9438:X
9416:2
9411:k
9408:2
9397:X
9366:)
9363:2
9359:/
9355:1
9352:(
9340:X
9318:2
9313:2
9302:X
9291:)
9287:(
9273:k
9260:X
9236:2
9231:k
9220:X
9209:)
9192:)
9189:c
9186:2
9183:=
9177:,
9174:2
9170:/
9163:=
9160:k
9157:(
9148:X
9145:c
9124:0
9118:c
9095:2
9079:X
9068:)
9063:k
9059:k
9039:2
9034:k
9021:2
9013:)
9010:1
9007:,
9004:0
9001:(
8996:k
8993:,
8987:,
8984:1
8981:=
8978:i
8973:N
8941:2
8936:1
8910:Y
8897:2
8884:=
8881:X
8860:)
8855:2
8847:,
8844:1
8841:(
8837:F
8830:Y
8802:2
8795:1
8765:Y
8760:1
8742:2
8729:=
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8706:)
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8693:,
8688:1
8680:(
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8669:Y
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8646:0
8643:=
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8613:0
8610:(
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8600:k
8580:2
8575:k
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8556:(
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5445:X
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5319:,
5310:0
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5299:N
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5280:1
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5266:,
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4860:Z
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4831:]
4816:1
4803:1
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4756:=
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4731:n
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4632:=
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4535:0
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4489:.
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4400:n
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4373:Z
4362:t
4358:Z
4354:(
4349:n
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4299:Z
4295:,
4292:.
4289:.
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4283:,
4278:1
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4217:.
4212:2
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3903:k
3899:;
3896:x
3893:(
3890:F
3868:2
3864:/
3860:x
3853:e
3847:2
3844:1
3839:=
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3833:2
3829:;
3826:x
3823:(
3820:f
3795:2
3791:/
3787:x
3780:e
3773:1
3770:=
3767:)
3764:2
3760:;
3757:x
3754:(
3751:F
3728:2
3725:=
3722:k
3695:)
3692:t
3689:,
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3680:P
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3653:t
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3647:s
3644:(
3618:,
3614:)
3608:2
3605:x
3599:,
3594:2
3591:k
3585:(
3581:P
3578:=
3572:)
3567:2
3564:k
3559:(
3551:)
3546:2
3543:x
3537:,
3532:2
3529:k
3524:(
3515:=
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3509:k
3505:;
3502:x
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3496:F
3477:)
3462:=
3459:k
3419:k
3395:k
3369:)
3366:2
3362:/
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3355:(
3322:.
3312:,
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3267:(
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3238:2
3234:/
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3207:/
3203:k
3199:x
3188:{
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3177:k
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3164:f
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3084:i
3080:p
3059:i
3039:i
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3009:N
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2997:E
2976:i
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2950:O
2927:2
2900:2
2866:i
2862:E
2855:2
2851:)
2845:i
2841:E
2832:i
2828:O
2824:(
2816:n
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2808:=
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2797:=
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2736:q
2733:N
2727:m
2721:N
2718:(
2712:+
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2703:N
2696:2
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2679:m
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2590:+
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2584:=
2581:N
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2558:p
2552:1
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2534:=
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2472:(
2466:=
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2424:1
2421:=
2418:q
2398:p
2378:N
2358:m
2331:q
2328:p
2325:N
2319:p
2316:N
2310:m
2304:=
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2243:1
2226:Q
2203:Q
2181:2
2177:Z
2173:=
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2150:)
2147:1
2144:,
2141:0
2138:(
2135:N
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2109:1
2089:0
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2025:t
2019:F
2012:t
1941:i
1937:Z
1929:k
1912:.
1907:2
1902:k
1885:Q
1865:)
1862:k
1859:(
1854:2
1837:Q
1824:k
1807:,
1802:2
1797:i
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1787:k
1782:1
1779:=
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1768:=
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1739:k
1735:Z
1731:1
1728:Z
1651:)
1648:k
1645:,
1640:2
1636:s
1632:(
1627:1
1622:W
1614:X
1594:)
1589:2
1585:s
1581:2
1578:=
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1567:2
1564:k
1559:=
1553:(
1542:X
1520:2
1515:k
1505:2
1501:s
1494:X
1464:2
1459:k
1449:2
1445:s
1418:)
1415:k
1412:,
1409:1
1406:(
1401:1
1396:W
1388:X
1328:)
1325:2
1322:=
1316:,
1311:2
1308:k
1303:=
1297:(
1286:X
1264:2
1259:k
1248:X
1218:2
1213:k
1179:k
1156:k
1132:2
1082:e
1074:t
1068:0
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1054:/
1050:k
1043:)
1039:t
1030:2
1024:1
1021:(
991:2
987:/
983:k
976:)
972:t
969:i
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960:1
957:(
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923:1
915:t
905:2
901:/
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890:)
886:t
883:2
877:1
874:(
841:)
836:2
833:k
828:(
820:)
814:2
811:k
803:1
799:(
795:+
783:)
777:)
770:2
767:k
760:(
752:2
748:(
738:+
729:2
726:k
690:k
654:k
650:/
646:8
615:k
612:2
583:)
580:0
577:,
574:2
568:k
565:(
531:3
525:)
516:k
513:9
509:2
501:1
496:(
491:k
460:k
430:)
424:2
421:x
415:,
410:2
407:k
401:(
390:)
387:2
383:/
379:k
376:(
369:1
336:2
332:/
328:x
321:e
315:1
309:2
305:/
301:k
297:x
289:)
286:2
282:/
278:k
275:(
267:2
263:/
259:k
255:2
250:1
219:)
213:+
210:,
207:0
204:[
198:x
156:N
148:k
117:2
112:k
86:)
83:k
80:(
75:2
27:.
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