//from http://www.kocw.net/home/search/kemView.do?kemId=1162312 5장_표본분포_중심극한의정리
[[확률변수,random_variable]]들 X,,1,,, …, X,,n,, 들이 독립이고 [[정규분포,normal_distribution]] N(μ, σ^^2^^)에 따른다면,
 확률변수들의 평균인 [[표본평균,sample_mean]]:
  $\bar{X}=\frac1{n}\sum_{i=1}^{n}X_i\sim N\left(\mu,\frac{\sigma^2}{n}\right)$
 표본평균 $\bar{X}$ 를 표준화한 확률변수:
  $Z=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1)$

'''중심극한정리'''
X,,1,,, …, X,,n,, : 독립, 유한평균 μ, 유한분산 σ^^2^^을 갖는 동일한 분포 →
 n이 커짐에 따라, $X_n=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1)$

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[[이항분포,binomial_distribution]]에서 [[정규분포,normal_distribution]]로의 근사화

Let $X_i\sim B(1,p)\quad(i=1,\cdots,n)$ : i.i.d (서로 독립)
$X=X_1+X_2+\cdots+X_n = \sum_{i=1}^{n}X_i$ :
 이항분포 B(n,p), 평균 μ=np, 분산 σ^^2^^=np(1-p)
By CLT,
 $\bar{X}=\frac1{n}\sum_{i=1}^{n}X_i \sim N\left(p,\frac{p(1-p)}{n}\right)$

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[[표본,sample]] [[표본평균,sample_mean]] [[정규분포,normal_distribution]] [[근사,approximation]] [[적률생성함수,moment_generating_function,MGF]]
[[모집단,population]]에서 뽑은 [[표본,sample]]이 충분히 크다면, [[표본평균,sample_mean]]의 분포는 [[정규분포,normal_distribution]]에 근사한다는 것
from/see http://blog.naver.com/mykepzzang/220851280035

= Related =
CLT와 ICA([[독립성분분석,independent_component_analysis,ICA]])은 반대 개념이라고... (see [[주성분분석,principal_component_analysis,PCA]])


= tmp links ko =
https://1992jhlee.tistory.com/12 for intuition

= tmp links en =
https://www.probabilitycourse.com/chapter7/7_1_2_central_limit_theorem.php

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[[WpSimple:Central_limit_theorem]]
[[https://terms.naver.com/entry.naver?docId=3405331&cid=47324&categoryId=47324 수학백과: 중심극한정리]]
[[WpEn:Central_limit_theorem]]
[[WpKo:중심_극한_정리]]
https://mathworld.wolfram.com/CentralLimitTheorem.html
https://encyclopediaofmath.org/wiki/Central_limit_theorem

https://everything2.com/title/Central+Limit+Theorem

Up: [[통계,statistics]]