# 大數法則

## 舉例

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

## 表現形式

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

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

### 弱大數定律

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

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

### 強大數定律

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

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

### 切比雪夫定理的特殊情況

${\displaystyle a_{1},\ a_{2},\ \dots \ ,\ a_{n},\ \dots }$ 為相互獨立的隨機變量，其數學期望為：${\displaystyle \operatorname {E} (a_{i})=\mu \quad (i=1,\ 2,\ \dots )}$方差為：${\displaystyle \operatorname {Var} (a_{i})=\sigma ^{2}\quad (i=1,\ 2,\ \dots )}$

### 伯努利大數定律

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

## 參考文獻

1. ^ Rick Durrett. Probability: Theory and Examples. Cambridge University Press. 2010: 61 [2013-11-18]. ISBN 978-0-521-76539-8 （英語）.