# 复合泊松分布

## 定义

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

$X_1, X_2, X_3, \dots$

$Y | N=\sum_{n=1}^N X_n$

## 性质

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

$\operatorname{Var}_Y(Y) = \operatorname{E}_N(N)\operatorname{Var}_X(X) + \left(\operatorname{E}_X(X)\right)^2\operatorname{Var}_N(N) .$

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

Y的概率分布可以由其特征函数决定：

$\varphi_Y(t) = \operatorname{E}\left(e^{itY}\right)= \operatorname{E}_N\left( \left(\operatorname{E}\left(e^{itX}\right) \right)^{N} \right)= \operatorname{E}_N\left( \left(\varphi_X(t) \right)^{N} \right), \,$

$\varphi_Y(t) = \textrm{e}^{\lambda(\varphi_X(t) - 1)}.\,$

## 复合泊松过程

$Y(t) = \sum_{i=0}^{N(t)} D_i$