Tag Archives: CLT

Central Limit Theorem

The central limit theorem is a fundamental theorem of statistics. It prescribes that the sum of a sufficiently large number of independent and identically distributed random variables approximately follows a normal distribution.

History of the Central Limit Theorem

The term “central limit theorem” most likely traces back to Georg Pólya. As he recapitulated at the beginning of a paper published in 1920, it was “generally known that the appearance of the Gaussian probability density ” in a great many situations “can be explained by one and the same limit theorem” which plays “a central role in probability theory”. Laplace had discovered the essentials of this fundamental theorem in 1810, and with the designation “central limit theorem of probability theory” which was even emphasized in the paper’s title, Pólya gave it the name that has been in general use ever since.

In this paper of 1820, Laplace starts by proving the central limit theorem for some certain probability distributions. He then continues with arbitrary discrete and continuous distributions. But a more general (and rigorous) proof should be attributed to Siméon Denis Poisson. He also intuited that weaker version could easily be derived. As for Laplace, the main purpose of that Central Limit Theorem for Poisson was to be a tool in calculations, not so much to be a mathematical theorem in itself. Therefore, neither Laplace nor Poisson explicitly formulate any conditions for the theorem to hold. The mathematical formulation of the theorem is due to the St. Petersburg School of probability, from 1870 until 1910, with Chebyshev, Markov and Liapounov.

Mathematical Formulation

Let be independent random variables that are identically distributed, with mean  and finite variance. Let

then from the law of large numbers  tend to 0 as  tends to infinity. The central limit theorem establishes that the distribution of  tends to a centered normal distribution when  goes to infinity. More specificaly,

We can also write

or .

A limiting result as an approximation

This central limit thereom is used to approximate distributions derived from summing, or averaging, identical random variables.

Consider for instance a course where 7 students out of 8 pass. What is the probability that (at least) 4 failed in a class of 25 students. Let be the dichotomous variable that describe failure : 1 if the student failed and 0 if he passed. That random variable has a Bernoulli distribution with parameter=1/8, with mean1/8 and variance 7/64. Consequently, if students’ grades are independent, the sum  follows a binomial distribution, with mean  and variance , which can be approximated, by the central limit theorem, by a normal distribution with mean  and variance . Here, =3.125 while =2.734. To compute  either either the binomial distribution, or the Gaussian approximation. In the first case, the probability is 80.47 %,

In the second case, use a continuity correction, and compute the probability that  is less than 4+1/2. From the central limit theorem

The probability that a standard Gaussian variable is less than this quantity is

which can be compared with 80.47% obtained without the approximation, see Figure 1. Note that this approximation was obtained by De Moivre, in 1713, and is usually known as « Bernoulli’s law of large numbers ».

Figure 1: Gaussian approximation of the binomial distribution.

Asymtptotic Confidence Intervals

The intuition is that a confidence interval is an interval in which one may be confident that a parameterof interest lies. For instance, that some quantity is measured , but the measurement is subject to a normally distributed error, with known variance . If  has a distribution, we know that

Equivalently, we could write


Thus, if   is measured to be , then the 95 % confidence interval for  is .

In the context of Bernoulli trials (described above), the asyymptotic 95 % confidence interval for p is

A popular rule of thumb can be derived when ~50%. In that context is close to 1.96 (or 2), and a 95 % approximated confidence interval is then

see Figure 2. If that confidence interval provides a good approximation for the 95 % confidence interval when ~50 %, it is an over-estimation when p is either much smaller, or much larger.

Figure 2: law of large numbers on the left, with the convergence of towards  as  increases, and central limit theorem, on the right, with the convergence of  towards a Gaussian distribution. The red area is the 95% confidence region.

The Delta Method and Method of Moments

This method is used to approximate a general transformation of a parameter that is known to be asymptotically normal,


Consider now a parametric model, … independent, with identical distribution (which can be a Weibull distribution to model a duration, a Pareto distribution to model the income or the wealth, etc). The method of moments is a method of estimating parameters based on equating population and sample values of certain moments of the distribution. For instance, if , then the estimator of the unknown parameter is given by equation or equivalently . From the central limit theorem

and applying the delta-method with , then

where a numerical approximation for the variance can be derived.This method has a long history, and has been intensively studied. Furthermore, this asymptotic normality can be used to compute a confidence interval, and also to derive an asymptotic testing procedure.

An Asymptotic Testing Procedure

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sthe>Eue="70241m-1   nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;Areime=nnbsp;nnbsp;(27)thos sth> t5os sthe>Eue="70241m-1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;Hivernnbsp;nnbsp;(e>)thos sth> t5os sthe>Eue="570833-1nnbsp;nnbsp;nnbsp;eCT2121nnbsp;nnbsp;(26)thos sth> t5os sthe>Eue="69m6ig1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n012nnbsp;nnbsp;(4)thos sth> t5os sthe>Eue="702541.1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n013nnbsp;nnbsp;(12)thos sth> t5os sthe>Eue="570811.1nnbsp;nnbsp;nnbsp;eCT6420nnbsp;nnbsp;(49)thos sth> t5os sthe>Eue="64735m-1   nnbsp;nnbsp;nnbsp;n012-Annbsp;nnbsp;(>)thos sth> t5os sthe>Eue="64735m-1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n012-Hnnbsp;nnbsp;(7)thos sth> t5os sthe>Eue="97350m-1   nnbsp;nnbsp;nnbsp;n014-Ennbsp;nnbsp;(9)thos sth> t5os sthe>Eue="62311m-1nnbsp;nnbsp;nnbsp;CIRMnnbsp;nnbsp;(6)thos sth> t5os sthe>Eue="623121">nnbsp;nnbsp;nnbsp;Crssh 48u arrnnbsp;nnbsp;(27)thos sth> t5os sthe>Eue="64713ht:nnbsp;nnbsp;nnbsp;ECO431nnbsp;nnbsp;(7)thos sth> t5os sthe>Eue="647130/1   ECO550nnbsp;nnbsp;(1)thos sth> t5os sthe>Eue="64713ig1nnbsp;nnbsp;nnbsp;ECO556nnbsp;nnbsp;(1)thos sth> t5os sthe>Eue="62311m-1nnbsp;nnbsp;nnbsp;ECO568nnbsp;nnbsp;(7)thos sth> t5os sthe>Eue="149754m-1nnbsp;nnbsp;nnbsp;onpaennbsp;nnbsp;(46)thos sth> t5os sthe>Eue="12m81t3-1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n015-n016nnbsp;nnbsp;(16)thos sth> t5os sthe>Eue="149757ht:nnbsp;nnbsp;nnbsp;ENSAEtnoul11;ivesuip>ertionoviennbsp;nnbsp;(29)thos sth> t5os sthe>Eue="1497577"1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n015-n016nnbsp;nnbsp;(2)thos sth> t5os sthe>Eue="1703t6/"1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n016-n017nnbsp;nnbsp;(e4)thos sth> t5os sthe>Eue="1703t6ig1nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n017-n018nnbsp;nnbsp;(13)thos sth> t5os sthe>Eue="973551">nnbsp;nnbsp;nnbsp;IA-dat>A-dsuflynnbsp;nnbsp;(11)thos sth> t5os sthe>Eue="62311ig1nnbsp;nnbsp;nnbsp;M1-a sutulltnnbsp;nnbsp;(7)thos sth> t5os sthe>Eue="131371m-1   M1-Dat> sbsuflynnbsp;nnbsp;(27)thos sth> t5os sthe>Eue="62311m->nnbsp;nnbsp;nnbsp;M1-économéoulennbsp;nnbsp;(1)thos sth> t5os sthe>Eue="123268ig1nnbsp;nnbsp;nnbsp;M1-se/tistsatinnbsp;nnbsp;(46)thos sth> t5os sthe>Eue="131371m-1nnbsp;nnbsp;nnbsp;M1-Trk" S ar t5os sthe>Eue="62311ht:nnbsp;nnbsp;nnbsp;M2-a sutulltnnbsp;nnbsp;(3)thos sth> t5os sthe>Eue="623113-1nnbsp;nnbsp;nnbsp;M2-ADDnnbsp;nnbsp;(e4)thos sth> t5os sthe>Eue="64717ht:nnbsp;nnbsp;nnbsp;M2-économéoule > ar iièosnnbsp;nnbsp;(3)thos sth> t5os sthe>Eue="1313717"1nnbsp;nnbsp;nnbsp;M2-iae socrtr)thos sth> t5os sthe>Eue="123268m-1nnbsp;nnbsp;nnbsp;M2-se/tennbsp;nnbsp;(1)thos sth> t5os sthe>Eue="70259/"1nnbsp;nnbsp;nnbsp;MAT8181nnbsp;nnbsp;(21)thos sth> t5os sthe>Eue="70259ig1nnbsp;nnbsp;nnbsp;MAT8595nnbsp;nnbsp;(12)thos sth> t5os sthe>Eue="623111">nnbsp;nnbsp;nnbsp;MAT8886nnbsp;nnbsp;(21)thos sth> t5os sthe>Eue="212070/1   networks  (9)thos sth> t5os sthe>Eue="l208500/1   nnbsp;nnbsp;nnbsp;n017/n018nnbsp;nnbsp;(9)thos sth> t5os sthe>Eue="2143623-1nnbsp;nnbsp;nnbsp;networks-n017nnbsp;nnbsp;(e)thos sth> t5os sthe>Eue="663300-1nnbsp;nnbsp;nnbsp;PhD 48u arnnbsp;nnbsp;(4)thos sth> t5os sthe>Eue="l208510">nnbsp;nnbsp;nnbsp;se/tistsa nnbsp;nnbsp;(2)thos sth> t5os sthe>Eue="l208511">nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;nnbsp;n017/n018nnbsp;nnbsp;(2)thos sth> t5os sthe>Eue="62310m-1nnbsp;nnbsp;nnbsp;STT2700nnbsp;nnbsp;(19)thos sth> t5os sthe>Eue="62311/"1nnbsp;nnbsp;nnbsp;STT6705Vnnbsp;nnbsp;(1)thos sth> t5os sthe>Eue="121595-1nnbsp;nnbsp;nnbsp;Traiaiapnnbsp;nnbsp;(2)thos sth> t5os sthe>Eue="6458">E thd ia nnbsp;nnbsp;(129)thos sth> t5os sthe>Eue="64732m-1nnbsp;nnbsp;nnbsp;C. Tttnnbsp;nnbsp;(1)thos sth> t5os sthe>Eue="62309th1nnbsp;nnbsp;nnbsp;Decier veTyeorynnbsp;nnbsp;(7)thos sth> t5os sthe>Eue="62316/"1nnbsp;nnbsp;nnbsp;egory/e aaasnnbsp;nnbsp;(4)thos sth> t5os sthe>Eue="10024ig1nnbsp;nnbsp;nnbsp;F%er innnbsp;nnbsp;(46)thos sth> t5os sthe>Eue="647180/1nnbsp;nnbsp;nnbsp;Gam t5os sthe>Eue="8805ig1nnbsp;nnbsp;nnbsp;Insuip>ernnbsp;nnbsp;(27)thos sth> t5os sthe>Eue="623097"1nnbsp;nnbsp;nnbsp;Microegory/ia nnbsp;nnbsp;(10)thos sth> t5os sthe>Eue="64715m-1nnbsp;nnbsp;nnbsp;Risknnbsp;nnbsp;(1>)thos sth> t5os sthe>Eue="245-1Innc>netnnbsp;nnbsp;(283)thos sth> t5os sthe>Eue="208m-1nnbsp;nnbsp;nnbsp;Brenospec>rnnbsp;nnbsp;(2e0)thos sth> t5os sthe>Eue="84030/1nnbsp;nnbsp;nnbsp;Misng Waneontnnbsp;nnbsp;(59)thos sth> t5os sthe>Eue="171th1nnbsp;nnbsp;nnbsp;Tutt 1rnnbsp;nnbsp;(e4)thos sth> t5os sthe>Eue="570801">M"extnnbsp;nnbsp;(88)thos sth> t5os sthe>Eue="6473e/"1nnbsp;nnbsp;nnbsp;Algebr>  (1)thos sth> t5os sthe>Eue="64733m-1nnbsp;nnbsp;nnbsp;Gereitry  (8)thos sth> t5os sthe>Eue="6473e1">nnbsp;nnbsp;nnbsp;Logra & an;tindwdoxntnnbsp;nnbsp;(6)thos sth> t5os sthe>Eue="647395-1nnbsp;nnbsp;nnbsp;Num arcnnbsp;nnbsp;(2)thos sth> t5os sthe>Eue="6473e0/1nnbsp;nnbsp;nnbsp;Os smizptstinnbsp;nnbsp;(8)thos sth> t5os sthe>Eue="193-1Nonm> t5os sthe>Eue="5366/"1R fext">nnbsp;nnbsp;(13i)thos sth> t5os sthe>Eue="97346m-1nnbsp;nnbsp;nnbsp;B eks  (12)thos sth> t5os sthe>Eue="1830/1nnbsp;nnbsp;nnbsp;Cofch/confrnnbsp;nnbsp;( />)thos sth> t5os sthe>Eue="24m-1nnbsp;nnbsp;nnbsp;Pubch/ptstisnnbsp;nnbsp;(6>)thos sth> t5os sthe>Eue="647380/1nnbsp;nnbsp;nnbsp;R axiaptnnbsp;nnbsp;(8)thos sth> t5os sthe>Eue="64716m-1nnbsp;nnbsp;nnbsp;Semaerannbsp;nnbsp;(/>)thos sth> t5os sthe>Eue="3697"1Se/tistsa nnbsp;nnbsp;(5e0)thos sth> t5os sthe>Eue="6473e7"1nnbsp;nnbsp;nnbsp;Bayrrs>nnnbsp;nnbsp;(12)thos sth> t5os sthe>Eue="3537"1nnbsp;nnbsp;nnbsp;Clrk/tinnbsp;nnbsp;(20)thos sth> t5os sthe>Eue="62307m->nnbsp;nnbsp;nnbsp;Cdpulvennbsp;nnbsp;(4>)thos sth> t5os sthe>Eue="3744/1nnbsp;nnbsp;nnbsp;Dat>maeiapnnbsp;nnbsp;(>)thos sth> t5os sthe>Eue="57087m->nnbsp;nnbsp;nnbsp;Dat> t5os sthe>Eue="2066-1   Demen8aphy  (29)thos sth> t5os sthe>Eue="623077"1nnbsp;nnbsp;nnbsp;ExSitme V>Euennbsp;nnbsp;(3i)thos sth> t5os sthe>Eue="570890/1nnbsp;nnbsp;nnbsp;GLMnnbsp;nnbsp;(29)thos sth> t5os sthe>Eue="6473em-1nnbsp;nnbsp;nnbsp;Histndynnbsp;nnbsp;(1)thos sth> t5os sthe>Eue="64m-1nnbsp;nnbsp;nnbsp;Ifch/confnnbsp;nnbsp;(2i)thos sth> t5os sthe>Eue="6473em-1nnbsp;nnbsp;nnbsp;Mng ir ti/ti Aerlysrennbsp;nnbsp;(>)thos sth> t5os sthe>Eue="570880/1nnbsp;nnbsp;nnbsp;Drobemscrtynnbsp;nnbsp;(7>)thos sth> t5os sthe>Eue="647137"1nnbsp;nnbsp;nnbsp;Rrs ome=srnnbsp;nnbsp;(26)thos sth> t5os sthe>Eue="57084m-1nnbsp;nnbsp;nnbsp;R0ics/remunnbsp;nnbsp;(56)thos sth> t5os sthe>Eue="66657"1nnbsp;nnbsp;nnbsp;Sthn/g> nnbsp;nnbsp;(6)thos sth> t5os sthe>Eue="570885-1nnbsp;nnbsp;nnbsp;Spptsalnnbsp;nnbsp;(13)thos sth> t5os sthe>Eue="647330/1nnbsp;nnbsp;nnbsp;Surv ynnbsp;nnbsp;(2)thos sth> t5os sthe>Eue="64714m-1nnbsp;nnbsp;nnbsp;Tost nnbsp;nnbsp;(19)thos sth> t5os sthe>Eue="57084m-1nnbsp;nnbsp;nnbsp;Trk" S ar t5os sthe>Eue="62320/"1Se/tistsa nnbsp;nnbsp;(i)thos sth> t5os sthe>Eue="70246m-1Tt snnbsp;nnbsp;(73)thos sth> r nge() { if ( dsi down.os sths[ dsi down.seped-edIndex ]./>Eue > 0 ) { dsi down.ptu nsNode.nt mit(); } } dsi down.onch>nge = onCptCx>nge; })(); /* ]]> */ < /csscalehn/s_w8ap/i>Decemb a n017 t5exead> t5trg-t sthA-dope="ccl"edrtle="Monday">MTWTFSS t5tf e > t5trg-t std 48lspan="3- /cssooev">« Nov="ar wtd> tt5td 4lvesssoad/1nnbsp; tt5td 48lspan="3- /cssno c"u4lvesssoad/1nnbsp; t t5tbody> t5trg-t std 48lspan="m-e>5td>15td>25td>3="ar wtd> t4="ar wtd>5td>5="ar wtd>5td>65td>789 wtd>10thtd>-t115td>125td>13="td>14="ar wtd>5td /csstoday">15="td>165td>17-t1819 wtd>20thtd>5td>215td>225td>23="td>24="td>-t25="td>265td>275td>2829 wtd>30thtd>5td>31-t t5/t bh-r w gchivrr-=0.>chivr/1<

chivrrthenttt5ul> ttt5/l>Decemb a n017 t5/l> t5/l>Odeib a n017 t5/l> t5/l>Augnse n017 t5/l>July n017 t5/l>June n017 t5/l>M"y n017 t5/l>Apsil n017 t5/l>M">ch n017 t5/l>Febru/st n017 t5/l> t5/l>Decemb a n016 t5/l> t5/l>Odeib a n016 t5/l>)th/l> t5/l>Augnse n016 t5/l>July n016)th/l> t5/l>June n016 t5/l>M"y n016 t5/l>Apsil n016 t5/l>M">ch n016 t5/l>Febru/st n016)th/l> t5/l> t5/l>Decemb a n015="arnnbsp;(8)th/l> t5/l>Odeib a n015="arnnbsp;(16)th/l> t5/l>Augnse n015="arnnbsp;(7)th/l> t5/l>July n015="arnnbsp;(11)th/l> t5/l>June n015="arnnbsp;(>)th/l> t5/l>M"y n015="arnnbsp;(e0)th/l> t5/l>Apsil n015="arnnbsp;(9)th/l> t5/l>M">ch n015="arnnbsp;(/>)th/l> t5/l>Febru/st n015="arnnbsp;(13)th/l> t5/l>Decemb a n014="arnnbsp;(16)th/l> t5/l>Odeib a n014="arnnbsp;(29)th/l> t5/l>Augnse n014="arnnbsp;(17)th/l> t5/l>July n014="arnnbsp;(21)th/l> t5/l>June n014="arnnbsp;(19)th/l> t5/l>M"y n014="arnnbsp;(19)th/l> t5/l>Apsil n014="arnnbsp;(/>)th/l> t5/l>M">ch n014="arnnbsp;(/2)th/l> t5/l>Febru/st n014="arnnbsp;(/2)th/l> t5/l>Decemb a n013="arnnbsp;(30)th/l> t5/l>Odeib a n013="arnnbsp;(33)th/l> t5/l>Augnse n013="arnnbsp;(e4)th/l> t5/l>July n013="arnnbsp;(9)th/l> t5/l>June n013="arnnbsp;(12)th/l> t5/l>M"y n013="arnnbsp;(20)th/l> t5/l>Apsil n013="arnnbsp;(2i)th/l> t5/l>M">ch n013="arnnbsp;(2i)th/l> t5/l>Febru/st n013="arnnbsp;(2>)th/l> t5/l>Decemb a n012 t5/l>)th/l> t5/l>Odeib a n012 t5/l> t5/l>Augnse n012)th/l> t5/l>July n012 t5/l>June n012 t5/l>M"y n012 t5/l>Apsil n012 t5/l>M">ch n012 t5/l>Febru/st n012 t5/l> t5/l>Decemb a n011 t5/l> t5/l>Odeib a n011 t5/l>)th/l> t5/l>Augnse n011)th/l> t5/l>July n011 t5/l>June n011 t5/l>M"y n011 t5/l>Apsil n011 t5/l>M">ch n011)th/l> t5/l>Febru/st n011 t5/l> t5/l>Decemb a n010tha1nnbsp;(9)th/l> t5/l> t5/l>Odeib a n010tha1nnbsp;(1>)th/l> t5/l> t5/l>Augnse n010tha1nnbsp;(7)th/l> t5/l>July n010tha1nnbsp;(3)th/l> t5/l>June n010tha1nnbsp;(12)th/l> t5/l>M"y n010tha1nnbsp;(11)th/l> t5/l>Apsil n010tha1nnbsp;(9)th/l> t5/l>M">ch n010tha1nnbsp;(11)th/l> t5/l>Febru/st n010tha1nnbsp;(6)th/l> t5/l> t5/l>Decemb a n009 wa1nnbsp;(6)th/l> t5/l>)th/l> t5/l>Odeib a n009 wa1nnbsp;(e4)th/l> t5/l> t5/l>Augnse n009 wa1nnbsp;(7)th/l> t5/l>July n009 wa1nnbsp;(7)th/l> t5/l>June n009 wa1nnbsp;(7)th/l> t5/l>M"y n009 wa1nnbsp;(6)th/l> t5/l>Apsil n009 wa1nnbsp;(e0)th/l> t5/l>M">ch n009 wa1nnbsp;(e>)th/l> t5/l>Febru/st n009 wa1nnbsp;(e9)th/l> t5/l> t5/l>Decemb a n008 t5/l> t5/l>Odeib a n008)th/l> t5/l> t5/l>July n008 t5/l>June n008 t5/l>M"y n008 t5/l>Apsil n008 t5/l>M">ch n008 t5/l>Febru/st n008 t5/l> t5/l>Decemb a 2007 t5/l>June n007 t5/l>M"y n007 t5/l>M">ch n007 t tt5/ailsoaenilso /csseita-the>

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