# 条件概率分布

## 数学定义

### 离散条件分布

${\displaystyle \forall j\in {\mathcal {J}},\quad p_{Y\mid X}(j)=p_{Y}(j\mid X=i)=P(Y=j\mid X=i)={\frac {P(X=i,Y=j)}{P(X=i)}}.}$${\displaystyle P(X=i)>0}$

${\displaystyle \forall i\in {\mathcal {I}},\quad p_{X\mid Y}(i)=p_{X}(i\mid Y=j)=P(X=i\mid Y=j)={\frac {P(X=i,Y=j)}{P(Y=j)}}.}$${\displaystyle P(Y=j)>0}$

${\displaystyle P(X=i)=p_{i.}=\sum _{j\in {\mathcal {J}}}p_{ij}}$
${\displaystyle P(Y=j)=p_{.j}=\sum _{i\in {\mathcal {I}}}p_{ij}}$

${\displaystyle p_{Y\mid X}(j)=P(Y=j\mid X=i)={\frac {p_{ij}}{p_{i.}}}.}$${\displaystyle p_{i.}>0}$

${\displaystyle p_{X\mid Y}(i)={\frac {p_{ij}}{p_{.j}}}.}$${\displaystyle p_{.j}>0}$

### 连续条件分布

${\displaystyle f_{Y|X}(y|x)=f_{Y}(y\mid X=x)={\frac {f(x,y)}{f_{X}(x)}}.}$

${\displaystyle f_{X|Y}(x|y)=f_{X}(x\mid Y=y)={\frac {f(x,y)}{f_{Y}(y)}}.}$

## 条件分布和独立分布

${\displaystyle P(Y=y\mid X=x)=P(Y=y)=p_{Y}(y)}$

${\displaystyle P(Y=y,X=x)=P(Y=y)\cdot P(X=x).}$

${\displaystyle P(Y=y)={\frac {P(Y=y,X=x)}{P(X=x)}}=P(Y=y\mid X=x).}$

## 参考资料

• 赵衡秀. 《概率论与数理统计》. 清华大学出版社. 2005.