# 复合泊松分布

## 定义

${\displaystyle N\sim \operatorname {Poisson} (\lambda ),}$

${\displaystyle X_{1},X_{2},X_{3},\dots }$

${\displaystyle Y|N=\sum _{n=1}^{N}X_{n}}$

## 性质

${\displaystyle \operatorname {E} _{Y}(Y)=\operatorname {E} _{N}\left[\operatorname {E} _{Y|N}(Y)\right]=\operatorname {E} _{N}\left[N\operatorname {E} _{X}(X)\right]=\operatorname {E} _{N}(N)\operatorname {E} _{X}(X),}$
${\displaystyle \operatorname {Var} _{Y}(Y)=E_{N}\left[\operatorname {Var} _{Y|N}(Y)\right]+\operatorname {Var} _{N}\left[E_{Y|N}(Y)\right]=\operatorname {E} _{N}\left[N\operatorname {Var} _{X}(X)\right]+\operatorname {Var} _{N}\left[N\operatorname {E} _{X}(X)\right]),}$

${\displaystyle \operatorname {Var} _{Y}(Y)=\operatorname {E} _{N}(N)\operatorname {Var} _{X}(X)+\left(\operatorname {E} _{X}(X)\right)^{2}\operatorname {Var} _{N}(N).}$

${\displaystyle \operatorname {E} (Y)=\operatorname {E} (N)\operatorname {E} (X),}$
${\displaystyle \operatorname {Var} (Y)=E(N)(\operatorname {Var} (X)+{E(X)}^{2})=E(N){E(X^{2})}.}$

Y的概率分布可以由其特征函数决定：

${\displaystyle \varphi _{Y}(t)=\operatorname {E} \left(e^{itY}\right)=\operatorname {E} _{N}\left(\left(\operatorname {E} \left(e^{itX}\right)\right)^{N}\right)=\operatorname {E} _{N}\left(\left(\varphi _{X}(t)\right)^{N}\right),\,}$

${\displaystyle \varphi _{Y}(t)={\textrm {e}}^{\lambda (\varphi _{X}(t)-1)}.\,}$

## 复合泊松过程

${\displaystyle Y(t)=\sum _{i=0}^{N(t)}D_{i}}$