Question

If X and Y are independent Poisson variates with mean lambda 1 and lambda 2 respectively, find the probability that x+y=k

Answer 1

Sums of independent Poisson random variables are Poisson random variables. Let $X$ and $Y$ be independent Poisson random variables with parameters $λ_1$ and $λ_2,$ respectively. Define $λ = λ_1 + λ_2$ and $Z = X + Y.$ We claim that $Z$ is a Poisson random variable with parameter $λ.$

Then

$$P(X+Y=k)=P(Z=k)=\dfrac{e^{-\lambda}(\lambda)^k}{k!}$$

$$P(X+Y=k)=\dfrac{e^{-\lambda_1+\lambda_2}(\lambda_1+\lambda_2)^k}{k!}$$