中心极限定理举例

1，独立同分布的中心极限定理：若 $$X_1,X_2,\cdots,X_n,\cdots$$ 为独立同分布的随机变量序列，$$E(X_i)=\mu, D(X_i)=\sigma^2, i=1,2,\cdots$$，则

$Y_n=\frac{\sum_{i=1}^nX_i-n\mu}{\sqrt{n}\sigma}$

$\lim_{n\to\infty}P\left\{\frac{\sum_{i=1}^nX_i-n\mu}{\sqrt{n}\sigma}\le x\right\}=\Phi(x)$

2，李雅普诺夫中心极限定理：设 $$X_1,X_2,\cdots,X_n,\cdots$$ 为随机变量序列，各自具有期望 $$E(X_i)=\mu_i, D(X_i)=\sigma_i^2$$，则

$Y_n=\frac{\sum_{i=1}^nX_i-\sum_{i=1}^n\mu_i}{\sqrt{\sum_{i=1}^n\sigma_i^2}}$

$\lim_{n\to\infty}P\left\{\frac{\sum_{i=1}^nX_i-\sum_{i=1}^n\mu_i}{\sqrt{\sum_{i=1}^n\sigma_i^2}}\le x\right\}=\Phi(x)$

3，棣莫弗-拉普拉斯中心极限定理：设 $$X_n$$ 为服从二项式分布 $$X\sim B(n,p)$$ 的随机变量序列，则

$Y_n=\frac{X_n-np}{\sqrt{np(1-p)}}$

$\lim_{n\to\infty}P\left\{\frac{X_n-np}{\sqrt{np(1-p)}}\le x\right\}=\Phi(x)$

$\lim_{n\to\infty}P\left\{\frac{\frac{1}{n}X_n-p}{\sqrt{\frac{p(1-p)}{n}}}\le x\right\}=\Phi(x)$

$E(X_i)=\frac{1}{6}(1+2+3+4+5+6)=\frac{7}{2}$

\begin{align*}D(X_i)&=\frac{1}{6}(1^2+2^2+3^2+4^2+5^2+6^2)-\left(\frac{7}{2}\right)^2\\ &=\frac{91}{6}-\frac{49}{4}=\frac{35}{12}\end{align*}

$E(\bar{X})=\frac{1}{100}\sum_{i=1}^{100}E(X_i)=\frac{7}{2}$

\begin{align*}D(\bar{X})&=D\left(\frac{1}{100}\sum_{i=1}^{100}D(X_i)\right)\\ &=\frac{1}{100^2}\cdot 100\cdot\frac{35}{12}=\frac{35}{1200}\end{align*}

\begin{align*}P(3\le \bar{X}\le 4)&=P\left\{\frac{3-\frac{7}{2}}{\sqrt{35/1200}}\le \frac{\bar{X}-\frac{7}{2}}{\sqrt{35/1200}}\le\frac{4-\frac{7}{2}}{\sqrt{35/1200}}\right\}\\ &=P\left\{\frac{-0.5\cdot\sqrt{1200}}{\sqrt{35}}\le \frac{\bar{X}-\frac{7}{2}}{\sqrt{35/1200}}\le\frac{0.5\cdot\sqrt{1200}}{\sqrt{35}}\right\}\\ &=P\left\{\frac{-10\sqrt{3}}{\sqrt{35}}\le \frac{\bar{X}-\frac{7}{2}}{\sqrt{35/1200}}\le\frac{10\sqrt{3}}{\sqrt{35}}\right\}\\ &=P\left\{-2.93\le \frac{\bar{X}-\frac{7}{2}}{\sqrt{35/1200}}\le2.93\right\}\\ &\approx \Phi(2.93)-\Phi(-2.93)=2\Phi(2.93)-1=0.9966\end{align*}

\begin{align*}P\left\{Y_n\ge 85\right\}&=P\left\{\frac{Y_n-90}{3}\ge \frac{85-90}{3}\right\}\\ &=1-P\left\{\frac{Y_n-90}{3}\le \frac{85-90}{3}\right\}\\ &=1-P\left\{\frac{Y_n-90}{3}\le \frac{-5}{3}\right\}\\ &=1-P\left\{\frac{Y_n-90}{3}\le -1.67\right\}\\ &\approx 1-\Phi(-1.67)=\Phi(1.67)=0.9525\end{align*}