Question

L.9) Central Limit Theorem Central Limit Theorem Version 1 says Go with independent random variables (Xi, X2, X3, ..., Xs, ...] all with the same cumulative distribution function so that μ-Expect[X] = Expect[X] and σ. varpKJ-Var[X] for all i and j Put As n gets large, the cumulative distribution function of S[n] is well approximated by the Normal[0, 1] cumulative distribution function. Another version of the Central Limit Theorem used often in statistics says Go with independent random variables (Xi. X. X3. X. ...J all with the same cumulative distribution function so that μ = Expect[Xi]-Expect[x,] and σ. = Var[X]=Val%) for all i and j Put SampleAverage[n] = As n gets large, the cumulative distribution function of Sample verage[n] is well approximated by the Normal Explain how the second version is a direct consequence of the first version. cumulative distribution function.

Answer 1

Centralmit Theo rem (CLT) orge ample from given a Butficienty o popudamon oih a finite levl of Varance, th mean ot al o be appro imatelyequal o po puaren 0 Somples Prom e Aame population oiabler o whe.е Ф (v) a 51ancloud Nolnict/ C mx.lane

As n gets large,fw Cumulative dirty bron function of sin」 well approni mahed bt tne Normal [0,1] Cumulanve. dlisri bution -funchon CcDr) Anohes Lesston Variable {t、h,A3_Nm.y rondom Independent Cutenulan ve effect of a dala , day x 4 heb randone a number f smal

So, by Central tinnit Theouem in fact is this principle, behind u fact that majority of measurable Variable are lomally disthe buted Becaue,an measurable characrertanc Can be Conteived f phenomenon ohich a hue Unde lying as a CLTtu mecuurable chasactenshic dlist ri buted Po erample, apenil an be maginede a ncl le all nuie Amaile penet