#### 期刊菜单

Conditional Independence of Random Events
DOI: 10.12677/AAM.2019.87139, PDF, HTML, XML, 下载: 1,043  浏览: 2,103  科研立项经费支持

Abstract: The concept and properties of condition independence of random events are introduced. The rela-tionship between independence of random events and conditional independence is discussed with examples, which shows that independence of random events and conditional independence do not imply each other. Finally, the determining theorems of conditional independence are provided.

1. 引言

1) 通过简单例子说明独立性和条件独立性是互不蕴含的关系。

2) 给出条件独立的判定条件。

2. 主要结果

2.1. 独立性和条件独立性

$P\left(A\cap B\right)=P\left(A\right)P\left(B\right)$ (1)

$P\left(A\cap B|C\right)=P\left(A|C\right)P\left(B|C\right)$ (2)

${B}_{1}$ = {第一次取得红球}， ${B}_{2}$ = {第二次取得红球}，

C = {第一次和第二次取得小球的颜色不同}，

R表示红球，W表示白球，则这个随机试验的样本空间为 $S=\left\{RW,WR,RR,WW\right\}$ ，且四种结果是等可能的。

$\begin{array}{l}P\left({B}_{1}\right)=\frac{1}{2},P\left({B}_{2}\right)=\frac{1}{2},P\left({B}_{1}\cap {B}_{2}\right)=\frac{1}{4},\\ P\left({B}_{1}|C\right)=\frac{1}{2},P\left({B}_{2}|C\right)=\frac{1}{2},P\left({B}_{1}\cap {B}_{2}|C\right)=0\end{array}$

$P\left({B}_{1}\cap {B}_{2}|C\right)\ne P\left({B}_{1}|C\right)P\left({B}_{2}|C\right)$ .

$A=\left\{1,2,3,4\right\},\text{\hspace{0.17em}}B=\left\{2,3,5,7\right\},\text{\hspace{0.17em}}C=\left\{2,3,7\right\}$ .

$\begin{array}{l}P\left(A\right)=\frac{4}{9},P\left(B\right)=\frac{4}{9},P\left(A\cap B\right)=\frac{2}{9},\\ P\left(A|C\right)=\frac{2}{3},P\left(B|C\right)=1,\text{\hspace{0.17em}}P\left(A\cap B|C\right)=\frac{2}{3}\end{array}$

$\begin{array}{l}P\left(A\cap B\right)\ne P\left(A\right)P\left(B\right),\\ P\left(A\cap B|C\right)=P\left(A|C\right)P\left(B|C\right)\end{array}$

2.2. 条件独立性的判定

$P\left(A|B\cap C\right)=P\left(A|C\right)$ (3)

3. 总结

NOTES

*通讯作者。

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