Central Limit Theorem

#Probability

2022/12/04

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Table of Content

• Theorem (central limit theorem)
  • Remark
  • De Moivre-Laplace Approximation to the Binomial

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Let \(X_1, \cdots X_n\) be independent, identically distributed r.v.s with finite mean \(\mu\) and variance \(\sigma^2\).

Let \(S_n = X_1 + \cdots + X_n\). Then mean of \(S_n\) is \(\mu\) and its variance is \(n\sigma^2\). To standardize it, we let

\[Z_n = {S_n - n\mu \over \sqrt{n}\sigma},\]

where

\[E[Z_n]=0, var(Z_n) = 1.\]

Theorem (central limit theorem)

Let \(Z\sim N(0, 1)\). Then for every \(z\), we have

\[\lim_{n\to \infty}P(Z_n\le z) = P(Z\le z).\]

Remark

• 把 \(Z_n\) 當 \(N(0, 1)\)，把 \(S_n\) 當 \(N(n\mu, n\sigma^2)\)。

• 對稱且 unimodal（單峰）的 \(X_i\) 近似效果最好。

  • 像是 uniform 和 binomial。

De Moivre-Laplace Approximation to the Binomial

待補。總之就是左右取 \({1\over 2}\)。