1401:
1396:
1391:
1386:
3646:
121:, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the twentieth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician
3570:
3979:
tables to format in limited pages; however, that sample size might be too small. See below "Using graphics and simulation.." by
Marasinghe et al, and see "Identification of Misconceptions in the Central Limit Theorem and Related Concepts and Evaluation of Computer Media as a Remedial
2863:
of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions tends to the characteristic function of the normal density as the number of
2855:
of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases
3980:
Tool" by Yu, Chong Ho and Dr. John T. Behrens, Arizona State
University & Spencer Anthony, Univ. of Oklahoma, Annual Meeting of the American Educational Research Association, presented April 19, 1995, paper revised in Feb 12, 1997, webpage (accessed 2007-10-25):
618:
113:, who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician
3971:> 29; however, research since 1990, has indicated larger samples, such as 100 or 250, might be needed if the population is skewed far from normal: the more skew, the larger the sample needed. The conditions might be rare, but critical when they occur:
1075:
1609:
The
Central Limit Theorem, as an approximation for a finite number of observations, provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
2895:
Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable (see
Rempala 2002).
1857:
3135:
2295:
3389:
1284:
38:, following a Gaussian distribution, or bell-shaped curve). The CLT indicates for large sample size (n>29 or 100), that the sampling distribution will have the same mean as the population, but variance divided by sample size (see:
3996:
Marasinghe, M., Meeker, W., Cook, D. & Shin, T.S.(1994 August), "Using graphics and simulation to teach statistical concepts", Paper presented at the Annual meeting of the
American Statistician Association, Toronto,
876:
2041:
495:
725:
3932:
2883:
of a product is simply the sum of the logs of the factors, so the log of a product of random variables that take only positive values tends to have a normal distribution, which makes the product itself have a
2741:
2345:" The idea is that dividing the function by appropriate normalizing functions and looking at the limiting behavior of the result can tell us much about the limiting behavior of the original function itself.
2448:
506:
370:
3208:
3696:
3016:
3863:
3700:
2564:
3357:
2395:
955:
2646:
3688:
2808:
67:, this explains the high frequency occurrence of the normal probability distribution. For other generalizations for finite variance which do not require identical distribution, see
2343:
2118:
2678:
1717:
2510:
1895:
1649:
independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a gaussian curve as
3787:
2832:
2765:
3027:
3565:{\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}{\mbox{E}}\left({\frac {(X_{i}-\mu _{i})^{2}}{s_{n}^{2}}}:\left|X_{i}-\mu _{i}\right|>\varepsilon s_{n}\right)=0}
1671:
2183:
1922:
1360:
2468:
2176:
2147:
2072:
1951:
57:
1095:
3807:
3711:
There are a number of useful and interesting examples arising from the central limit theorem. Below are brief outlines of two such examples and here are a
4030:, Nauchnoe Nasledie P.L.Chebysheva. Vypusk Pervyi: Matematika. (Russian) Edited by S. N. Bernstein.] Academiya Nauk SSSR, Moscow-Leningrad, 1945. 174 pp.
125:
defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial
784:
2767:
is the largest power of n which if serves as a normalizing function would provide a non-trivial (non-zero) limiting behavior. Interestingly enough,
2771:
tells us what is happening "in between" The Law of Large
Numbers and The Central Limit Theorem. Specifically it says that the normalizing function
4097:
3755:. Due to the central limit theorem this smoothing can be approximated by several filter steps that can be computed much faster, like the simple
1958:
109:
The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born
English mathematician
1289:
But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the
3734:
Flipping a large number of coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).
1677:
article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
2860:
751:
420:
629:
613:{\displaystyle \lim _{n\rightarrow \infty }{\mbox{P}}\left({\frac {{\overline {X}}_{n}-\mu }{\sigma /{\sqrt {n}}}}\leq z\right)=\Phi (z)}
3868:
2888:. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the product of different
2879:
The central limit theorem tells us what to expect about the sum of independent random variables, but what about the product? Well, the
2683:
2400:
305:
1409:
40:
19:
4073:
3716:
3379:
In the same setting and with the same notation as above, we can replace the
Lyapunov condition with the following weaker one (from
3150:
2947:
3812:
3712:
1617:, although in this case the convergence of the sum toward a normal distribution has singular properties: namely, a sum of
2905:
2517:
3293:
2360:
1417:
A graphical representation of the centra limit theorem can be formed by plotting random means of a population. Consider
1070:{\displaystyle Z_{n}={\frac {n{\overline {X}}_{n}-n\mu }{\sigma {\sqrt {n}}}}=\sum _{i=1}^{n}{Y_{i} \over {\sqrt {n}}}.}
2149:
and its approximation and then dividing by the next term in the expansion we arrive to a more refined statement about
3667:
2592:
2357:
is being studied in classical probability theory. Under certain regularity conditions, by The Law of Large
Numbers,
3692:
1688:
as well as The
Central Limit Theorem are partial solutions to a general problem: "What is the limiting behavior of
404:
755:
4056:
4019:
S. Artstein, K. Ball, F. Barthe and A. Naor, "Solution of Shannon's Problem on the Monotonicity of Entropy",
2768:
86:
4081:
2774:
2848:
1634:
1374:
82:
49:
4067:
2302:
2077:
1852:{\displaystyle f(n)=a_{1}\varphi _{1}(n)+a_{2}\varphi _{2}(n)+O(\varphi _{3}(n))\ (n\rightarrow \infty ).}
2299:
here one can say that: "the difference between the function and its approximation grows approximately as
1566:
For the CLT, it is recommended to plot the means upwards to 30 points (sample size 30).If we standardize
396:
252:
2651:
2473:
1290:
3943:
1864:
3687:
There are some theorems which treat the case of sums of non-independent variables, for instance the
1630:
1622:
1618:
1614:
195:
188:
30:(CLT) states that if the sum of the variables has a finite variance, then it will be approximately
4051:
3765:
3380:
3130:{\displaystyle r_{n}^{3}=\sum _{i=1}^{n}{\mbox{E}}\left({\left|X_{i}-\mu _{i}\right|}^{3}\right)}
2920:
be a sequence of independent random variables defined on the same probability space. Assume that
2885:
2813:
2746:
69:
2290:{\displaystyle \lim _{n\to \infty }{\frac {f(n)-a_{1}\varphi _{1}(n)}{\varphi _{2}(n)}}=a_{2}}
1674:
1656:
1924:- the coefficient at the highest-order term in the expansion representing the rate at which
1900:
1685:
1626:
1338:
1325:
1306:
1279:{\displaystyle \left^{n}=\left^{n}\,\rightarrow \,e^{-t^{2}/2},\quad n\rightarrow \infty .}
114:
2453:
2152:
2123:
2048:
1927:
8:
4105:
is a site with many resources for teaching statistics including the Central Limit Theorem
4061:
3728:
1638:
1332:
1317:
775:
747:
388:
269:
31:
4077:
3972:
3792:
3214:
1363:
203:
147:
122:
76:
53:
3752:
2868:
1702:
184:
139:
110:
1366:
to that of the normal distribution, as proven by Artstein, Ball, Barthe and Naor.
4087:
3744:
1080:
By simple properties of characteristic functions, the characteristic function of
60:
will tend to be distributed according to a particular "attractor distribution".
3756:
890:
871:{\displaystyle \varphi _{Y}(t)=1-{t^{2} \over 2}+o(t^{2}),\quad t\rightarrow 0}
199:
2864:
density functions increases without bound, under the conditions stated above.
158:
The theorem most often called the central limit theorem is the following. Let
3981:
143:
4080:(Select the Sampling Distribution CLT Experiment from the drop-down list of
3759:. From the central limit theorem you know, that for achieving a Gaussian of
1613:
The Central Limit theorem also applies to sums of independent and identical
3975:
are used to illustrate the cases. The cutoff with n > 29 has allowed
2036:{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{\varphi _{1}(n)}}=a_{1}.}
4091:
3748:
3724:
2871:, since the characteristic function is essentially a Fourier transform.
2852:
1378:
1294:
731:
408:
384:
138:
See Bernstein (1945) for a historical discussion focusing on the work of
4012:
2834:
of The Central Limit Theorem provides a non-trivial limiting behavior.
743:
1705:
is one of the most popular tools employed to approach such questions.
1400:
1395:
1390:
1385:
3976:
2880:
1642:
1370:
490:{\displaystyle \lim _{n\to \infty }{\mbox{P}}(Z_{n}\leq z)=\Phi (z),}
126:
3371:
converges towards the standard normal distribution N(0,1) as above.
4033:
G. Rempala and J. Wesolowski, "Asymptotics of products of sums and
3760:
2512:
which provide values of first two constants in informal expansion:
928:, it is easy to see that the standardised mean of the observations
720:{\displaystyle {\overline {X}}_{n}=S_{n}/n=(X_{1}+\cdots +X_{n})/n}
287:
In order to clarify the word "approaches" in the last sentence, we
180:
64:
3927:{\displaystyle \sigma ^{2}=\sigma _{1}^{2}+\dots +\sigma _{n}^{2}}
3623:
and zero otherwise. Then the distribution of the standardized sum
3665:
parameter to this template to explain the issue with the article.
2736:{\displaystyle {\frac {S_{n}-n\mu }{n^{\frac {1}{\beta }}}}\to 0}
1297:
of characteristic functions implies convergence in distribution.
288:
150:
that led to the first proofs of the C.L.T. in a general setting.
2443:{\displaystyle {\frac {S_{n}-n\mu }{\sqrt {n}}}\rightarrow \xi }
1316:− μ)) exists and is finite, then the above convergence is
750:, the central limit theorem has a remarkably simple proof using
365:{\displaystyle Z_{n}={\frac {S_{n}-n\mu }{\sigma {\sqrt {n}}}}.}
2889:
2810:
intermediate in size between n of The Law of Large Numbers and
3723:
The probability distribution for total distance covered in a
3203:{\displaystyle \lim _{n\to \infty }{\frac {r_{n}}{s_{n}}}=0.}
4070:
interactive simulation w/ a variety of modifiable parameters
4064:
interactive simulation to experiment with various parameters
1606:
as above, and it approaches a standard normal distribution.
63:
Since many real populations yield distributions with finite
3632:
converges towards the standard normal distribution N(0,1).
1331:
The convergence normal is monotonic, in the sense that the
1320:
and the speed of convergence is at least on the order of 1/
763:
35:
4102:
2348:
Informally, something along these lines is happening when
3011:{\displaystyle s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}.}
4013:
Understanding Probability: Chance Rules in Everyday Life
2874:
2837:
259:. Furthermore, informally speaking, the distribution of
58:
independent and identically-distributed random variables
2851:
of the sum of two or more independent variables is the
1680:
3858:{\displaystyle \sigma _{1}^{2},\dots ,\sigma _{n}^{2}}
3431:
3071:
527:
441:
3871:
3815:
3795:
3768:
3392:
3296:
3153:
3030:
2950:
2816:
2777:
2749:
2686:
2654:
2595:
2520:
2476:
2456:
2403:
2363:
2305:
2186:
2155:
2126:
2080:
2051:
1961:
1930:
1903:
1867:
1720:
1659:
1442:
represents a single random variable from the sample:
1341:
1098:
958:
787:
737:
632:
509:
423:
308:
2892:factors, so they follow a log-normal distribution.
2559:{\displaystyle S_{n}\approx \mu n+\xi {\sqrt {n}}.}
1637:corresponding to a continuous variable (namely the
1633:whose probability distribution converges towards a
1473:. N represents the size of the population. Derive
1369:Pictures of a distribution being "smoothed out" by
4028:On the work of P.L.Chebyshev in Probability Theory
3926:
3857:
3801:
3781:
3564:
3352:{\displaystyle Z_{n}={\frac {S_{n}-m_{n}}{s_{n}}}}
3351:
3202:
3129:
3010:
2856:without bound, under the conditions stated above.
2826:
2802:
2759:
2735:
2672:
2640:
2558:
2504:
2462:
2442:
2390:{\displaystyle {\frac {S_{n}}{n}}\rightarrow \mu }
2389:
2337:
2289:
2170:
2141:
2112:
2066:
2035:
1945:
1916:
1889:
1851:
1665:
1354:
1278:
1069:
870:
719:
612:
489:
364:
183:of random variables which are defined on the same
153:
3599:}), i.e., the expectation of the random variable
117:rescued it from obscurity in his monumental work
56:. They all express the fact that any sum of many
4068:CLT in NetLogo (Connected Probability - ProbLab)
3394:
3155:
2188:
1963:
1701:approaches infinity?" In mathematical analysis,
742:For a theorem of such fundamental importance to
511:
425:
1435:will represent the mean of a random sample and
4016:, Cambridge: Cambridge University Press, 2004.
2641:{\displaystyle E(|X_{1}|)^{\beta }<\infty }
3655:needs attention from an expert in Mathematics
1377:and three subsequent summations, obtained by
4021:Journal of the American Mathematical Society
3992:
3990:
3706:
1708:Suppose we have an asymptotic expansion of
1300:
897: that goes to zero more rapidly than
3967:For decades, large sample size was set as
3697:central limit theorem for mixing processes
2867:An equivalent statement can be made about
754:. It is similar to the proof of a (weak)
3987:
3751:of a signal with an appropriately scaled
1410:Illustration of the central limit theorem
1231:
1227:
20:Illustration of the central limit theorem
4088:Generate sampling distributions in Excel
4039:Electronic Communications in Probability
3727:(biased or unbiased) will tend toward a
919:− μ)/σ, the standardised value of
3963:
3961:
3959:
3701:central limit theorem for convex bodies
3635:
770:) = 1), the characteristic function of
4074:General Central Limit Theorem Activity
3743:Signals can be smoothed by applying a
3670:may be able to help recruit an expert.
3374:
3217:condition). We again consider the sum
3021:Assume that the third central moments
2803:{\displaystyle {\sqrt {n\log \log n}}}
2899:
2875:Products of positive random variables
2838:Alternative statements of the theorem
1412:for further details on these images.)
3956:
3738:
3639:
2842:
2338:{\displaystyle a_{2}\varphi _{2}(n)}
2113:{\displaystyle a_{1}\varphi _{1}(n)}
1681:Relation to the law of large numbers
102:
119:Théorie Analytique des Probabilités
13:
3809:filters with windows of variances
3404:
3165:
2635:
2397:and by The Central Limit Theorem,
2198:
1973:
1897:and taking the limit will produce
1840:
1660:
1645:of the realisations of the sum of
1270:
738:Proof of the central limit theorem
598:
521:
472:
435:
14:
4115:
4045:
3689:m-dependent central limit theorem
2769:The Law of the Iterated Logarithm
2673:{\displaystyle 1\leq \beta <2}
2120:". Taking the difference between
1641:). This means that if we build a
1625:, so that we are confronted to a
1482:from 1 to whichever sample size.
3713:large number of CLT applications
3693:martingale central limit theorem
3644:
2906:Lyapunov's central limit theorem
2505:{\displaystyle N(0,\sigma ^{2})}
1399:
1394:
1389:
1384:
405:cumulative distribution function
2935:and finite standard deviation σ
1890:{\displaystyle \varphi _{1}(n)}
1597:), we obtain the same variable
1263:
858:
154:Classical central limit theorem
4090:Specify arbitrary population,
3472:
3445:
3401:
3269:and its standard deviation is
3162:
2727:
2623:
2618:
2603:
2599:
2499:
2480:
2434:
2381:
2332:
2326:
2268:
2262:
2247:
2241:
2215:
2209:
2195:
2165:
2159:
2136:
2130:
2107:
2101:
2061:
2055:
2011:
2005:
1990:
1984:
1970:
1953:changes in its leading term.
1940:
1934:
1884:
1878:
1843:
1837:
1831:
1825:
1822:
1816:
1803:
1794:
1788:
1762:
1756:
1730:
1724:
1267:
1228:
862:
852:
839:
804:
798:
706:
674:
607:
601:
518:
481:
475:
466:
447:
432:
1:
4004:
3383:in 1920). For every ε > 0
238:. Then the expected value of
229: + ... +
4052:Animated examples of the CLT
1635:probability density function
984:
758:. For any random variable,
639:
547:
389:standard normal distribution
100:Tijms (2004, p.169) writes:
7:
3937:
3782:{\displaystyle \sigma ^{2}}
3715:, presented as part of the
2929:has finite expected value μ
2827:{\displaystyle {\sqrt {n}}}
2760:{\displaystyle {\sqrt {n}}}
397:convergence in distribution
395:approaches ∞ (this is
10:
4120:
4057:Central Limit Theorem Java
4041:, vol. 7, pp. 47-54, 2002.
2568:It could be shown that if
2045:Informally, one can say: "
1293:, which confirms that the
407:of N(0,1), then for every
95:
3707:Applications and examples
3362:then the distribution of
1631:discrete random variables
1619:discrete random variables
1615:discrete random variables
28:The Central Limit Theorem
3949:
3235:. The expected value of
1686:The law of large numbers
1623:discrete random variable
1301:Convergence to the limit
752:characteristic functions
189:probability distribution
4094:, and sample statistic.
3668:WikiProject Mathematics
2886:log-normal distribution
2861:characteristic function
2589:, ... are i.i.d. and
2074:grows approximately as
1861:dividing both parts by
1666:{\displaystyle \infty }
1381:of density functions):
1375:density of distribution
1291:Lévy continuity theorem
893:" for some function of
766:and unit variance (var(
210:exist and are finite.
198:. Assume that both the
129:of probability theory.
3928:
3859:
3803:
3783:
3566:
3429:
3353:
3204:
3131:
3069:
3012:
2989:
2828:
2804:
2761:
2737:
2674:
2642:
2560:
2506:
2464:
2444:
2391:
2339:
2291:
2172:
2143:
2114:
2068:
2037:
1947:
1918:
1891:
1853:
1667:
1356:
1280:
1071:
1042:
872:
721:
614:
491:
375:Then, distribution of
366:
4062:Central Limit Theorem
3929:
3860:
3804:
3784:
3567:
3409:
3354:
3205:
3140:are finite for every
3132:
3049:
3013:
2969:
2829:
2805:
2762:
2738:
2675:
2643:
2561:
2507:
2465:
2445:
2392:
2340:
2292:
2173:
2144:
2115:
2069:
2038:
1948:
1919:
1917:{\displaystyle a_{1}}
1892:
1854:
1675:binomial distribution
1668:
1357:
1355:{\displaystyle Z_{n}}
1305:If the third central
1281:
1072:
1022:
873:
722:
615:
492:
367:
46:central limit theorem
4076:& corresponding
3869:
3813:
3793:
3766:
3747:, which is just the
3636:Non-independent case
3390:
3294:
3278:. If we standardize
3151:
3028:
2948:
2814:
2775:
2747:
2684:
2652:
2593:
2518:
2474:
2463:{\displaystyle \xi }
2454:
2401:
2361:
2303:
2184:
2171:{\displaystyle f(n)}
2153:
2142:{\displaystyle f(n)}
2124:
2078:
2067:{\displaystyle f(n)}
2049:
1959:
1946:{\displaystyle f(n)}
1928:
1901:
1865:
1718:
1657:
1339:
1326:Berry-Esséen theorem
1096:
956:
785:
756:law of large numbers
630:
507:
421:
399:). This means: if Φ(
306:
115:Pierre-Simon Laplace
32:normally distributed
4023:17, 975-982 (2004).
3973:computer animations
3923:
3899:
3854:
3830:
3729:normal distribution
3496:
3375:Lindeberg condition
3045:
3004:
2965:
1639:normal distribution
748:applied probability
270:normal distribution
48:is any of a set of
41:Illustration of CLT
3924:
3909:
3885:
3855:
3840:
3816:
3799:
3789:you have to apply
3779:
3562:
3482:
3435:
3408:
3349:
3200:
3169:
3127:
3075:
3031:
3008:
2990:
2951:
2900:Lyapunov condition
2869:Fourier transforms
2824:
2800:
2757:
2733:
2670:
2638:
2556:
2502:
2470:is distributed as
2460:
2440:
2387:
2335:
2287:
2202:
2168:
2139:
2110:
2064:
2033:
1977:
1943:
1914:
1887:
1849:
1663:
1373:(showing original
1352:
1276:
1067:
868:
717:
610:
531:
525:
500:or, equivalently,
487:
445:
439:
362:
204:standard deviation
148:Aleksandr Lyapunov
123:Aleksandr Lyapunov
54:probability theory
3802:{\displaystyle n}
3753:Gaussian function
3739:Signal processing
3717:SOCR CLT Activity
3685:
3684:
3611:} whose value is
3497:
3434:
3393:
3347:
3192:
3154:
3074:
2843:Density functions
2822:
2798:
2755:
2725:
2722:
2551:
2432:
2431:
2379:
2272:
2187:
2015:
1962:
1830:
1703:asymptotic series
1210:
1183:
1131:
1129:
1062:
1060:
1017:
1014:
987:
891:little o notation
831:
642:
582:
579:
550:
530:
510:
444:
424:
357:
354:
213:Consider the sum
187:, share the same
185:probability space
142:and his students
140:Pafnuty Chebyshev
136:
135:
111:Abraham de Moivre
4111:
4082:SOCR Experiments
3998:
3994:
3985:
3965:
3933:
3931:
3930:
3925:
3922:
3917:
3898:
3893:
3881:
3880:
3864:
3862:
3861:
3856:
3853:
3848:
3829:
3824:
3808:
3806:
3805:
3800:
3788:
3786:
3785:
3780:
3778:
3777:
3680:
3677:
3671:
3657:. Please add a
3648:
3647:
3640:
3571:
3569:
3568:
3563:
3555:
3551:
3550:
3549:
3534:
3530:
3529:
3528:
3516:
3515:
3498:
3495:
3490:
3481:
3480:
3479:
3470:
3469:
3457:
3456:
3443:
3436:
3432:
3428:
3423:
3407:
3358:
3356:
3355:
3350:
3348:
3346:
3345:
3336:
3335:
3334:
3322:
3321:
3311:
3306:
3305:
3209:
3207:
3206:
3201:
3193:
3191:
3190:
3181:
3180:
3171:
3168:
3136:
3134:
3133:
3128:
3126:
3122:
3121:
3116:
3115:
3111:
3110:
3109:
3097:
3096:
3076:
3072:
3068:
3063:
3044:
3039:
3017:
3015:
3014:
3009:
3003:
2998:
2988:
2983:
2964:
2959:
2833:
2831:
2830:
2825:
2823:
2818:
2809:
2807:
2806:
2801:
2799:
2779:
2766:
2764:
2763:
2758:
2756:
2751:
2742:
2740:
2739:
2734:
2726:
2724:
2723:
2715:
2709:
2699:
2698:
2688:
2679:
2677:
2676:
2671:
2647:
2645:
2644:
2639:
2631:
2630:
2621:
2616:
2615:
2606:
2565:
2563:
2562:
2557:
2552:
2547:
2530:
2529:
2511:
2509:
2508:
2503:
2498:
2497:
2469:
2467:
2466:
2461:
2449:
2447:
2446:
2441:
2433:
2427:
2426:
2416:
2415:
2405:
2396:
2394:
2393:
2388:
2380:
2375:
2374:
2365:
2344:
2342:
2341:
2336:
2325:
2324:
2315:
2314:
2296:
2294:
2293:
2288:
2286:
2285:
2273:
2271:
2261:
2260:
2250:
2240:
2239:
2230:
2229:
2204:
2201:
2177:
2175:
2174:
2169:
2148:
2146:
2145:
2140:
2119:
2117:
2116:
2111:
2100:
2099:
2090:
2089:
2073:
2071:
2070:
2065:
2042:
2040:
2039:
2034:
2029:
2028:
2016:
2014:
2004:
2003:
1993:
1979:
1976:
1952:
1950:
1949:
1944:
1923:
1921:
1920:
1915:
1913:
1912:
1896:
1894:
1893:
1888:
1877:
1876:
1858:
1856:
1855:
1850:
1828:
1815:
1814:
1787:
1786:
1777:
1776:
1755:
1754:
1745:
1744:
1672:
1670:
1669:
1664:
1403:
1398:
1393:
1388:
1361:
1359:
1358:
1353:
1351:
1350:
1285:
1283:
1282:
1277:
1259:
1258:
1254:
1249:
1248:
1226:
1225:
1220:
1216:
1215:
1211:
1206:
1205:
1196:
1184:
1182:
1174:
1173:
1164:
1147:
1146:
1141:
1137:
1136:
1132:
1130:
1125:
1120:
1114:
1113:
1076:
1074:
1073:
1068:
1063:
1061:
1056:
1054:
1053:
1044:
1041:
1036:
1018:
1016:
1015:
1010:
1004:
994:
993:
988:
980:
973:
968:
967:
877:
875:
874:
869:
851:
850:
832:
827:
826:
817:
797:
796:
776:Taylor's theorem
726:
724:
723:
718:
713:
705:
704:
686:
685:
667:
662:
661:
649:
648:
643:
635:
619:
617:
616:
611:
594:
590:
583:
581:
580:
575:
573:
564:
557:
556:
551:
543:
539:
532:
528:
524:
496:
494:
493:
488:
459:
458:
446:
442:
438:
371:
369:
368:
363:
358:
356:
355:
350:
344:
334:
333:
323:
318:
317:
103:
50:weak-convergence
4119:
4118:
4114:
4113:
4112:
4110:
4109:
4108:
4078:SOCR CLT Applet
4048:
4026:S.N.Bernstein,
4007:
4002:
4001:
3995:
3988:
3966:
3957:
3952:
3944:Diversification
3940:
3918:
3913:
3894:
3889:
3876:
3872:
3870:
3867:
3866:
3849:
3844:
3825:
3820:
3814:
3811:
3810:
3794:
3791:
3790:
3773:
3769:
3767:
3764:
3763:
3745:Gaussian filter
3741:
3709:
3681:
3675:
3672:
3666:
3649:
3645:
3638:
3631:
3545:
3541:
3524:
3520:
3511:
3507:
3506:
3502:
3491:
3486:
3475:
3471:
3465:
3461:
3452:
3448:
3444:
3442:
3441:
3437:
3430:
3424:
3413:
3397:
3391:
3388:
3387:
3377:
3370:
3341:
3337:
3330:
3326:
3317:
3313:
3312:
3310:
3301:
3297:
3295:
3292:
3291:
3286:
3277:
3268:
3262:
3252:
3243:
3233:
3229:
3222:
3186:
3182:
3176:
3172:
3170:
3158:
3152:
3149:
3148:
3117:
3105:
3101:
3092:
3088:
3087:
3083:
3082:
3081:
3077:
3070:
3064:
3053:
3040:
3035:
3029:
3026:
3025:
2999:
2994:
2984:
2973:
2960:
2955:
2949:
2946:
2945:
2940:
2934:
2928:
2919:
2902:
2877:
2845:
2840:
2817:
2815:
2812:
2811:
2778:
2776:
2773:
2772:
2750:
2748:
2745:
2744:
2714:
2710:
2694:
2690:
2689:
2687:
2685:
2682:
2681:
2653:
2650:
2649:
2626:
2622:
2617:
2611:
2607:
2602:
2594:
2591:
2590:
2588:
2581:
2574:
2546:
2525:
2521:
2519:
2516:
2515:
2493:
2489:
2475:
2472:
2471:
2455:
2452:
2451:
2411:
2407:
2406:
2404:
2402:
2399:
2398:
2370:
2366:
2364:
2362:
2359:
2358:
2356:
2320:
2316:
2310:
2306:
2304:
2301:
2300:
2281:
2277:
2256:
2252:
2251:
2235:
2231:
2225:
2221:
2205:
2203:
2191:
2185:
2182:
2181:
2154:
2151:
2150:
2125:
2122:
2121:
2095:
2091:
2085:
2081:
2079:
2076:
2075:
2050:
2047:
2046:
2024:
2020:
1999:
1995:
1994:
1980:
1978:
1966:
1960:
1957:
1956:
1929:
1926:
1925:
1908:
1904:
1902:
1899:
1898:
1872:
1868:
1866:
1863:
1862:
1810:
1806:
1782:
1778:
1772:
1768:
1750:
1746:
1740:
1736:
1719:
1716:
1715:
1696:
1683:
1658:
1655:
1654:
1605:
1592:
1583:
1574:
1562:
1555:
1548:
1541:
1527:
1520:
1513:
1499:
1492:
1481:
1468:
1459:
1452:
1441:
1434:
1425:
1346:
1342:
1340:
1337:
1336:
1315:
1303:
1250:
1244:
1240:
1236:
1232:
1221:
1201:
1197:
1195:
1191:
1175:
1169:
1165:
1163:
1156:
1152:
1151:
1142:
1124:
1119:
1115:
1109:
1105:
1104:
1100:
1099:
1097:
1094:
1093:
1088:
1055:
1049:
1045:
1043:
1037:
1026:
1009:
1005:
989:
979:
978:
974:
972:
963:
959:
957:
954:
953:
947:
940:
933:
927:
918:
909:
846:
842:
822:
818:
816:
792:
788:
786:
783:
782:
740:
709:
700:
696:
681:
677:
663:
657:
653:
644:
634:
633:
631:
628:
627:
574:
569:
565:
552:
542:
541:
540:
538:
537:
533:
526:
514:
508:
505:
504:
454:
450:
440:
428:
422:
419:
418:
383:
349:
345:
329:
325:
324:
322:
313:
309:
307:
304:
303:
298:
268:approaches the
267:
246:
237:
228:
221:
178:
171:
164:
156:
98:
92:
44:). Formally, a
12:
11:
5:
4117:
4107:
4106:
4100:
4099:Another proof.
4095:
4085:
4071:
4065:
4059:
4054:
4047:
4046:External links
4044:
4043:
4042:
4037:-statistics",
4031:
4024:
4017:
4006:
4003:
4000:
3999:
3986:
3954:
3953:
3951:
3948:
3947:
3946:
3939:
3936:
3921:
3916:
3912:
3908:
3905:
3902:
3897:
3892:
3888:
3884:
3879:
3875:
3852:
3847:
3843:
3839:
3836:
3833:
3828:
3823:
3819:
3798:
3776:
3772:
3757:moving average
3740:
3737:
3736:
3735:
3732:
3708:
3705:
3683:
3682:
3652:
3650:
3643:
3637:
3634:
3627:
3573:
3572:
3561:
3558:
3554:
3548:
3544:
3540:
3537:
3533:
3527:
3523:
3519:
3514:
3510:
3505:
3501:
3494:
3489:
3485:
3478:
3474:
3468:
3464:
3460:
3455:
3451:
3447:
3440:
3427:
3422:
3419:
3416:
3412:
3406:
3403:
3400:
3396:
3376:
3373:
3366:
3360:
3359:
3344:
3340:
3333:
3329:
3325:
3320:
3316:
3309:
3304:
3300:
3282:
3273:
3264:
3254:
3248:
3239:
3231:
3227:
3220:
3211:
3210:
3199:
3196:
3189:
3185:
3179:
3175:
3167:
3164:
3161:
3157:
3138:
3137:
3125:
3120:
3114:
3108:
3104:
3100:
3095:
3091:
3086:
3080:
3067:
3062:
3059:
3056:
3052:
3048:
3043:
3038:
3034:
3019:
3018:
3007:
3002:
2997:
2993:
2987:
2982:
2979:
2976:
2972:
2968:
2963:
2958:
2954:
2936:
2930:
2924:
2915:
2901:
2898:
2876:
2873:
2844:
2841:
2839:
2836:
2821:
2797:
2794:
2791:
2788:
2785:
2782:
2754:
2732:
2729:
2721:
2718:
2713:
2708:
2705:
2702:
2697:
2693:
2669:
2666:
2663:
2660:
2657:
2637:
2634:
2629:
2625:
2620:
2614:
2610:
2605:
2601:
2598:
2586:
2579:
2572:
2555:
2550:
2545:
2542:
2539:
2536:
2533:
2528:
2524:
2501:
2496:
2492:
2488:
2485:
2482:
2479:
2459:
2439:
2436:
2430:
2425:
2422:
2419:
2414:
2410:
2386:
2383:
2378:
2373:
2369:
2352:
2334:
2331:
2328:
2323:
2319:
2313:
2309:
2284:
2280:
2276:
2270:
2267:
2264:
2259:
2255:
2249:
2246:
2243:
2238:
2234:
2228:
2224:
2220:
2217:
2214:
2211:
2208:
2200:
2197:
2194:
2190:
2167:
2164:
2161:
2158:
2138:
2135:
2132:
2129:
2109:
2106:
2103:
2098:
2094:
2088:
2084:
2063:
2060:
2057:
2054:
2032:
2027:
2023:
2019:
2013:
2010:
2007:
2002:
1998:
1992:
1989:
1986:
1983:
1975:
1972:
1969:
1965:
1942:
1939:
1936:
1933:
1911:
1907:
1886:
1883:
1880:
1875:
1871:
1848:
1845:
1842:
1839:
1836:
1833:
1827:
1824:
1821:
1818:
1813:
1809:
1805:
1802:
1799:
1796:
1793:
1790:
1785:
1781:
1775:
1771:
1767:
1764:
1761:
1758:
1753:
1749:
1743:
1739:
1735:
1732:
1729:
1726:
1723:
1692:
1682:
1679:
1662:
1601:
1588:
1579:
1570:
1560:
1553:
1546:
1537:
1525:
1518:
1509:
1497:
1488:
1477:
1464:
1457:
1448:
1439:
1430:
1421:
1415:
1414:
1349:
1345:
1313:
1302:
1299:
1287:
1286:
1275:
1272:
1269:
1266:
1262:
1257:
1253:
1247:
1243:
1239:
1235:
1230:
1224:
1219:
1214:
1209:
1204:
1200:
1194:
1190:
1187:
1181:
1178:
1172:
1168:
1162:
1159:
1155:
1150:
1145:
1140:
1135:
1128:
1123:
1118:
1112:
1108:
1103:
1084:
1078:
1077:
1066:
1059:
1052:
1048:
1040:
1035:
1032:
1029:
1025:
1021:
1013:
1008:
1003:
1000:
997:
992:
986:
983:
977:
971:
966:
962:
945:
938:
931:
923:
914:
905:
879:
878:
867:
864:
861:
857:
854:
849:
845:
841:
838:
835:
830:
825:
821:
815:
812:
809:
806:
803:
800:
795:
791:
739:
736:
728:
727:
716:
712:
708:
703:
699:
695:
692:
689:
684:
680:
676:
673:
670:
666:
660:
656:
652:
647:
641:
638:
621:
620:
609:
606:
603:
600:
597:
593:
589:
586:
578:
572:
568:
563:
560:
555:
549:
546:
536:
523:
520:
517:
513:
498:
497:
486:
483:
480:
477:
474:
471:
468:
465:
462:
457:
453:
449:
437:
434:
431:
427:
379:
373:
372:
361:
353:
348:
343:
340:
337:
332:
328:
321:
316:
312:
294:
284:approaches ∞.
263:
253:standard error
242:
233:
226:
217:
200:expected value
176:
169:
162:
155:
152:
134:
133:
130:
107:
97:
94:
25:
24:
9:
6:
4:
3:
2:
4116:
4104:
4101:
4098:
4096:
4093:
4089:
4086:
4083:
4079:
4075:
4072:
4069:
4066:
4063:
4060:
4058:
4055:
4053:
4050:
4049:
4040:
4036:
4032:
4029:
4025:
4022:
4018:
4015:
4014:
4009:
4008:
3993:
3991:
3983:
3978:
3974:
3970:
3964:
3962:
3960:
3955:
3945:
3942:
3941:
3935:
3919:
3914:
3910:
3906:
3903:
3900:
3895:
3890:
3886:
3882:
3877:
3873:
3850:
3845:
3841:
3837:
3834:
3831:
3826:
3821:
3817:
3796:
3774:
3770:
3762:
3758:
3754:
3750:
3746:
3733:
3730:
3726:
3722:
3721:
3720:
3718:
3714:
3704:
3702:
3698:
3694:
3690:
3679:
3669:
3664:
3660:
3656:
3653:This article
3651:
3642:
3641:
3633:
3630:
3626:
3622:
3618:
3614:
3610:
3606:
3602:
3598:
3594:
3590:
3586:
3582:
3578:
3559:
3556:
3552:
3546:
3542:
3538:
3535:
3531:
3525:
3521:
3517:
3512:
3508:
3503:
3499:
3492:
3487:
3483:
3476:
3466:
3462:
3458:
3453:
3449:
3438:
3425:
3420:
3417:
3414:
3410:
3398:
3386:
3385:
3384:
3382:
3372:
3369:
3365:
3342:
3338:
3331:
3327:
3323:
3318:
3314:
3307:
3302:
3298:
3290:
3289:
3288:
3285:
3281:
3276:
3272:
3267:
3261:
3257:
3251:
3247:
3242:
3238:
3234:
3223:
3216:
3213:(This is the
3197:
3194:
3187:
3183:
3177:
3173:
3159:
3147:
3146:
3145:
3143:
3123:
3118:
3112:
3106:
3102:
3098:
3093:
3089:
3084:
3078:
3065:
3060:
3057:
3054:
3050:
3046:
3041:
3036:
3032:
3024:
3023:
3022:
3005:
3000:
2995:
2991:
2985:
2980:
2977:
2974:
2970:
2966:
2961:
2956:
2952:
2944:
2943:
2942:
2939:
2933:
2927:
2923:
2918:
2914:
2909:
2907:
2897:
2893:
2891:
2887:
2882:
2872:
2870:
2865:
2862:
2857:
2854:
2850:
2835:
2819:
2795:
2792:
2789:
2786:
2783:
2780:
2770:
2752:
2730:
2719:
2716:
2711:
2706:
2703:
2700:
2695:
2691:
2667:
2664:
2661:
2658:
2655:
2632:
2627:
2612:
2608:
2596:
2585:
2578:
2571:
2566:
2553:
2548:
2543:
2540:
2537:
2534:
2531:
2526:
2522:
2513:
2494:
2490:
2486:
2483:
2477:
2457:
2437:
2428:
2423:
2420:
2417:
2412:
2408:
2384:
2376:
2371:
2367:
2355:
2351:
2346:
2329:
2321:
2317:
2311:
2307:
2297:
2282:
2278:
2274:
2265:
2257:
2253:
2244:
2236:
2232:
2226:
2222:
2218:
2212:
2206:
2192:
2179:
2162:
2156:
2133:
2127:
2104:
2096:
2092:
2086:
2082:
2058:
2052:
2043:
2030:
2025:
2021:
2017:
2008:
2000:
1996:
1987:
1981:
1967:
1954:
1937:
1931:
1909:
1905:
1881:
1873:
1869:
1859:
1846:
1834:
1819:
1811:
1807:
1800:
1797:
1791:
1783:
1779:
1773:
1769:
1765:
1759:
1751:
1747:
1741:
1737:
1733:
1727:
1721:
1713:
1711:
1706:
1704:
1700:
1695:
1691:
1687:
1678:
1676:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1611:
1607:
1604:
1600:
1596:
1591:
1587:
1582:
1578:
1573:
1569:
1564:
1559:
1552:
1545:
1540:
1536:
1532:
1531:
1524:
1517:
1512:
1508:
1504:
1503:
1496:
1491:
1487:
1483:
1480:
1476:
1472:
1467:
1463:
1456:
1451:
1447:
1443:
1438:
1433:
1429:
1424:
1420:
1413:
1411:
1406:
1405:
1404:
1402:
1397:
1392:
1387:
1382:
1380:
1376:
1372:
1367:
1365:
1364:monotonically
1347:
1343:
1334:
1329:
1327:
1323:
1319:
1312:
1308:
1298:
1296:
1292:
1273:
1264:
1260:
1255:
1251:
1245:
1241:
1237:
1233:
1222:
1217:
1212:
1207:
1202:
1198:
1192:
1188:
1185:
1179:
1176:
1170:
1166:
1160:
1157:
1153:
1148:
1143:
1138:
1133:
1126:
1121:
1116:
1110:
1106:
1101:
1092:
1091:
1090:
1087:
1083:
1064:
1057:
1050:
1046:
1038:
1033:
1030:
1027:
1023:
1019:
1011:
1006:
1001:
998:
995:
990:
981:
975:
969:
964:
960:
952:
951:
950:
948:
941:
934:
926:
922:
917:
913:
908:
904:
900:
896:
892:
888:
884:
865:
859:
855:
847:
843:
836:
833:
828:
823:
819:
813:
810:
807:
801:
793:
789:
781:
780:
779:
777:
773:
769:
765:
761:
757:
753:
749:
745:
735:
733:
714:
710:
701:
697:
693:
690:
687:
682:
678:
671:
668:
664:
658:
654:
650:
645:
636:
626:
625:
624:
604:
595:
591:
587:
584:
576:
570:
566:
561:
558:
553:
544:
534:
515:
503:
502:
501:
484:
478:
469:
463:
460:
455:
451:
429:
417:
416:
415:
413:
410:
406:
402:
398:
394:
390:
386:
382:
378:
359:
351:
346:
341:
338:
335:
330:
326:
319:
314:
310:
302:
301:
300:
297:
293:
290:
285:
283:
279:
275:
271:
266:
262:
258:
254:
250:
245:
241:
236:
232:
225:
222: =
220:
216:
211:
209:
205:
201:
197:
193:
190:
186:
182:
175:
168:
161:
151:
149:
145:
144:Andrey Markov
141:
131:
128:
124:
120:
116:
112:
108:
105:
104:
101:
93:
90:
88:
84:
80:
78:
73:
71:
66:
61:
59:
55:
51:
47:
43:
42:
37:
33:
29:
23:
21:
16:
15:
4103:CAUSEweb.org
4038:
4034:
4027:
4020:
4011:
4010:Henk Tijms,
3968:
3742:
3710:
3686:
3673:
3662:
3658:
3654:
3628:
3624:
3620:
3616:
3612:
3608:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3574:
3378:
3367:
3363:
3361:
3287:by setting
3283:
3279:
3274:
3270:
3265:
3259:
3255:
3249:
3245:
3240:
3236:
3225:
3218:
3212:
3141:
3139:
3020:
2941:. We define
2937:
2931:
2925:
2921:
2916:
2912:
2910:
2903:
2894:
2878:
2866:
2858:
2846:
2583:
2576:
2569:
2567:
2514:
2353:
2349:
2347:
2298:
2180:
2044:
1955:
1860:
1714:
1709:
1707:
1698:
1693:
1689:
1684:
1650:
1646:
1612:
1608:
1602:
1598:
1594:
1593:- μ) / (σ /
1589:
1585:
1580:
1576:
1571:
1567:
1565:
1557:
1550:
1543:
1538:
1534:
1533:
1529:
1522:
1515:
1510:
1506:
1505:
1501:
1494:
1489:
1485:
1484:
1478:
1474:
1470:
1465:
1461:
1454:
1449:
1445:
1444:
1436:
1431:
1427:
1422:
1418:
1416:
1407:
1383:
1368:
1330:
1321:
1310:
1304:
1288:
1085:
1081:
1079:
943:
936:
929:
924:
920:
915:
911:
906:
902:
898:
894:
889: ) is "
886:
882:
880:
771:
767:
762:, with zero
759:
741:
729:
622:
499:
411:
400:
392:
387:towards the
380:
376:
374:
299:by setting
295:
291:
286:
281:
277:
273:
264:
260:
256:
248:
243:
239:
234:
230:
223:
218:
214:
212:
207:
191:
173:
166:
159:
157:
137:
118:
99:
91:
75:
68:
62:
45:
39:
27:
26:
17:
4092:sample size
3982:CWisdom-rtf
3749:convolution
3725:random walk
3144:, and that
2853:convolution
1653:approaches
1621:is still a
1575:by setting
1379:convolution
1295:convergence
732:sample mean
409:real number
289:standardize
196:independent
179:, ... be a
52:results in
4005:References
3676:March 2009
2859:Since the
1362:increases
901:. Letting
744:statistics
414:, we have
391:N(0,1) as
255:is σ
251:μ and its
202:μ and the
87:Kolmogorov
18:Also see:
3977:Student-t
3911:σ
3904:⋯
3887:σ
3874:σ
3842:σ
3835:…
3818:σ
3771:σ
3575:where E(
3539:ε
3522:μ
3518:−
3463:μ
3459:−
3411:∑
3405:∞
3402:→
3381:Lindeberg
3324:−
3166:∞
3163:→
3103:μ
3099:−
3051:∑
2992:σ
2971:∑
2904:See also
2881:logarithm
2793:
2787:
2728:→
2720:β
2707:μ
2701:−
2662:β
2659:≤
2648:for some
2636:∞
2628:β
2544:ξ
2535:μ
2532:≈
2491:σ
2458:ξ
2438:ξ
2435:→
2424:μ
2418:−
2385:μ
2382:→
2318:φ
2254:φ
2233:φ
2219:−
2199:∞
2196:→
2093:φ
1997:φ
1974:∞
1971:→
1870:φ
1841:∞
1838:→
1808:φ
1780:φ
1748:φ
1661:∞
1643:histogram
1371:summation
1271:∞
1268:→
1238:−
1229:→
1161:−
1107:φ
1024:∑
1007:σ
1002:μ
996:−
985:¯
863:→
814:−
790:φ
691:⋯
640:¯
599:Φ
585:≤
567:σ
562:μ
559:−
548:¯
522:∞
519:→
473:Φ
461:≤
436:∞
433:→
403:) is the
385:converges
347:σ
342:μ
336:−
127:sovereign
79:condition
72:condition
70:Lindeberg
3938:See also
3761:variance
3699:and the
3587:) is E(
3579: :
3215:Lyapunov
1460:+ ... +
949:is just
194:and are
181:sequence
89:states.
83:Gnedenko
77:Lyapunov
65:variance
3997:Canada.
2849:density
1333:entropy
1318:uniform
942:, ...,
885: (
774:is, by
730:is the
96:History
3695:, the
3691:, the
3659:reason
3230:+...+X
2890:random
2743:hence
2450:where
1829:
1673:. The
1627:series
1307:moment
881:where
623:where
3950:Notes
3865:with
3661:or a
3619:>
3607:>
3595:>
3583:>
2680:then
1408:(See
1324:(see
280:) as
206:σ of
3663:talk
3536:>
3258:=1..
2911:Let
2847:The
2665:<
2633:<
1710:f(n)
1563:)/3
1500:) /
1469:) /
910:be (
764:mean
746:and
146:and
85:and
36:i.e.
3615:if
3395:lim
3253:= ∑
3244:is
3156:lim
2790:log
2784:log
2189:lim
1964:lim
1697:as
1629:of
1584:= (
1542:= (
1528:)/
1514:= (
1493:= (
1453:= (
1335:of
1328:).
1309:E((
1089:is
512:lim
426:lim
276:μ,σ
247:is
3989:^
3958:^
3934:.
3719:.
3703:.
3603:1{
3591:1{
3198:0.
2908:.
2582:,
2575:,
2178::
1712::
1556:+
1549:+
1521:+
1426:.
935:,
778:,
734:.
272:N(
172:,
165:,
132:”
106:“
81:,
74:,
4084:)
4035:U
3984:.
3969:n
3920:2
3915:n
3907:+
3901:+
3896:2
3891:1
3883:=
3878:2
3851:2
3846:n
3838:,
3832:,
3827:2
3822:1
3797:n
3775:2
3731:.
3678:)
3674:(
3629:n
3625:Z
3621:c
3617:V
3613:U
3609:c
3605:V
3601:U
3597:c
3593:V
3589:U
3585:c
3581:V
3577:U
3560:0
3557:=
3553:)
3547:n
3543:s
3532:|
3526:i
3513:i
3509:X
3504:|
3500::
3493:2
3488:n
3484:s
3477:2
3473:)
3467:i
3454:i
3450:X
3446:(
3439:(
3433:E
3426:n
3421:1
3418:=
3415:i
3399:n
3368:n
3364:Z
3343:n
3339:s
3332:n
3328:m
3319:n
3315:S
3308:=
3303:n
3299:Z
3284:n
3280:S
3275:n
3271:s
3266:i
3263:μ
3260:n
3256:i
3250:n
3246:m
3241:n
3237:S
3232:n
3228:1
3226:X
3224:=
3221:n
3219:S
3195:=
3188:n
3184:s
3178:n
3174:r
3160:n
3142:n
3124:)
3119:3
3113:|
3107:i
3094:i
3090:X
3085:|
3079:(
3073:E
3066:n
3061:1
3058:=
3055:i
3047:=
3042:3
3037:n
3033:r
3006:.
3001:2
2996:i
2986:n
2981:1
2978:=
2975:i
2967:=
2962:2
2957:n
2953:s
2938:n
2932:n
2926:n
2922:X
2917:n
2913:X
2820:n
2796:n
2781:n
2753:n
2731:0
2717:1
2712:n
2704:n
2696:n
2692:S
2668:2
2656:1
2624:)
2619:|
2613:1
2609:X
2604:|
2600:(
2597:E
2587:3
2584:X
2580:2
2577:X
2573:1
2570:X
2554:.
2549:n
2541:+
2538:n
2527:n
2523:S
2500:)
2495:2
2487:,
2484:0
2481:(
2478:N
2429:n
2421:n
2413:n
2409:S
2377:n
2372:n
2368:S
2354:n
2350:S
2333:)
2330:n
2327:(
2322:2
2312:2
2308:a
2283:2
2279:a
2275:=
2269:)
2266:n
2263:(
2258:2
2248:)
2245:n
2242:(
2237:1
2227:1
2223:a
2216:)
2213:n
2210:(
2207:f
2193:n
2166:)
2163:n
2160:(
2157:f
2137:)
2134:n
2131:(
2128:f
2108:)
2105:n
2102:(
2097:1
2087:1
2083:a
2062:)
2059:n
2056:(
2053:f
2031:.
2026:1
2022:a
2018:=
2012:)
2009:n
2006:(
2001:1
1991:)
1988:n
1985:(
1982:f
1968:n
1941:)
1938:n
1935:(
1932:f
1910:1
1906:a
1885:)
1882:n
1879:(
1874:1
1847:.
1844:)
1835:n
1832:(
1826:)
1823:)
1820:n
1817:(
1812:3
1804:(
1801:O
1798:+
1795:)
1792:n
1789:(
1784:2
1774:2
1770:a
1766:+
1763:)
1760:n
1757:(
1752:1
1742:1
1738:a
1734:=
1731:)
1728:n
1725:(
1722:f
1699:n
1694:n
1690:S
1651:n
1647:n
1603:n
1599:Z
1595:n
1590:n
1586:A
1581:n
1577:Z
1572:n
1568:A
1561:3
1558:X
1554:2
1551:X
1547:1
1544:X
1539:3
1535:A
1530:2
1526:2
1523:X
1519:1
1516:X
1511:2
1507:A
1502:1
1498:1
1495:X
1490:1
1486:A
1479:n
1475:A
1471:n
1466:n
1462:X
1458:1
1455:X
1450:n
1446:A
1440:n
1437:X
1432:n
1428:A
1423:n
1419:A
1348:n
1344:Z
1322:n
1314:1
1311:X
1274:.
1265:n
1261:,
1256:2
1252:/
1246:2
1242:t
1234:e
1223:n
1218:]
1213:)
1208:n
1203:2
1199:t
1193:(
1189:o
1186:+
1180:n
1177:2
1171:2
1167:t
1158:1
1154:[
1149:=
1144:n
1139:]
1134:)
1127:n
1122:t
1117:(
1111:Y
1102:[
1086:n
1082:Z
1065:.
1058:n
1051:i
1047:Y
1039:n
1034:1
1031:=
1028:i
1020:=
1012:n
999:n
991:n
982:X
976:n
970:=
965:n
961:Z
946:n
944:X
939:2
937:X
932:1
930:X
925:i
921:X
916:i
912:X
907:i
903:Y
899:t
895:t
887:t
883:o
866:0
860:t
856:,
853:)
848:2
844:t
840:(
837:o
834:+
829:2
824:2
820:t
811:1
808:=
805:)
802:t
799:(
794:Y
772:Y
768:Y
760:Y
715:n
711:/
707:)
702:n
698:X
694:+
688:+
683:1
679:X
675:(
672:=
669:n
665:/
659:n
655:S
651:=
646:n
637:X
608:)
605:z
602:(
596:=
592:)
588:z
577:n
571:/
554:n
545:X
535:(
529:P
516:n
485:,
482:)
479:z
476:(
470:=
467:)
464:z
456:n
452:Z
448:(
443:P
430:n
412:z
401:z
393:n
381:n
377:Z
360:.
352:n
339:n
331:n
327:S
320:=
315:n
311:Z
296:n
292:S
282:n
278:n
274:n
265:n
261:S
257:n
249:n
244:n
240:S
235:n
231:X
227:1
224:X
219:n
215:S
208:D
192:D
177:3
174:X
170:2
167:X
163:1
160:X
34:(
22:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.