独立泊松之和

Two independent Poisson

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Given two independent Poisson random variables $X$ and $Y$ with parameters $\lambda$ and $\mu$ respectively, what is the probability that the sum of the two variables is equal to $n$ ?

解析

若 $X\sim\mathrm{Poisson}(\lambda)$ 、 $Y\sim\mathrm{Poisson}(\mu)$ 独立，则

X+Y\sim\mathrm{Poisson}(\lambda+\mu).

因此

\boxed{\mathbb{P}(X+Y=n)=e^{-(\lambda+\mu)}\frac{(\lambda+\mu)^n}{n!}}.