# 独立 (概率论)

（重定向自独立 (统计)

## 獨立事件

### 对于任意個事件

${\displaystyle \mathrm {Pr} (A_{i}\cap A_{j})=\mathrm {Pr} (A_{i})\mathrm {Pr} (A_{j})}$

${\displaystyle \Pr(A_{1}\cap \cdots \cap A_{n})=\Pr(A_{1})\,\cdots \,\Pr(A_{n})}$

### 独立事件的性质

${\displaystyle \Pr(A\mid B)=\Pr(A)\,}$

${\displaystyle \Pr(A\mid B)={\Pr(A\cap B) \over \Pr(B)},}$  (只要Pr(B) ≠ 0 )

${\displaystyle \Pr(A\cap B)=\Pr(A)\Pr(B)}$

${\displaystyle \Pr(A)=\Pr(A\cap A)=\Pr(A)\Pr(A)\,}$

## 獨立隨機變量

XY是獨立的，則其期望值E會有下列的好性質： E[X Y] = E[X] E[Y], （假定都存在）且其方差（若存在）满足

var(X + Y) = var(X) + var(Y),

${\displaystyle F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),}$

${\displaystyle f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)}$

## 條件獨立隨機變數

${\displaystyle \mathrm {P} (X\leq x,Y\leq y\;|\;Z=z)=\mathrm {Pr} (X\leq x\;|\;Z=z)\cdot \mathrm {Pr} (Y\leq y\;|\;Z=z)}$

${\displaystyle p_{XY|Z}(x,y|z)=p_{X|Z}(x|z)\cdot p_{Y|Z}(y|z)}$

${\displaystyle \ \mathrm {Pr} (X=x|Y=y,Z=z)=\mathrm {Pr} (X=x|Z=z)}$

## 書籍

• Kirby Faciane (2006). Statistics for Empirical and Quantitative Finance. H.C. Baird: Philadelphia. ISBN 0-9788208-9-4.
1. ^ Feller, W. Stochastic Independence. An Introduction to Probability Theory and Its Applications. Wiley. 1971.