# 大数定律

（重定向自大数定律）

## 举例

${\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5}$

## 表现形式

${\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}$

${\displaystyle {\overline {X}}_{n}\to \mu \quad {\textrm {as}}\quad n\to \infty }$

### 弱大数定律

${\displaystyle {\overline {X}}_{n}\ {\xrightarrow {P}}\ \mu \quad {\textrm {as}}\quad n\to \infty }$

${\displaystyle \lim _{n\to \infty }P\left(\,|{\overline {X}}_{n}-\mu |>\varepsilon \,\right)=0}$

### 强大数定律

${\displaystyle {\overline {X}}_{n}\ {\xrightarrow {\text{a.s.}}}\ \mu \quad {\textrm {as}}\quad n\to \infty }$

${\displaystyle P\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1}$

### 切比雪夫定理的特殊情况

${\displaystyle a_{1},\ a_{2},\ \dots \ ,\ a_{n},\ \dots }$ 为相互独立的随机变量，其数学期望为：${\displaystyle \operatorname {E} (a_{i})=\mu \quad (i=1,\ 2,\ \dots )}$方差为：${\displaystyle \operatorname {Var} (a_{i})=\sigma ^{2}\quad (i=1,\ 2,\ \dots )}$

### 伯努利大数定律

${\displaystyle \lim _{n\to \infty }{P{\left\{\left|{\frac {n_{x}}{n}}-p\right|<\varepsilon \right\}}}=1}$