中心极限定理

设\(X_{1},\ldots ,X_{n},\ldots \)为独立同分布的随机变量序列，其均值为\(\mu\)，方差为\(\sigma^{2}\),则随机变量：

\begin{equation} \label{eq:1} \frac{X_{1} + \ldots + X_{n} - n\mu}{\sigma \sqrt{n}} \end{equation}

的分布当\(n\to \infty\)时趋向于标准正态分布。即对任何\(-\infty < a < \infty\)

\begin{equation} \label{eq:2} P\bigg\{ \frac{X_{1} + \ldots + X_{n} - n\mu}{\sigma\sqrt{n}} \leq a \bigg\} \to \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{a}e^{-x^{2}/2}dx \quad n\to \infty \end{equation}