Sum of Independent Poisson Random Variables is Poisson

Theorem
Let $X$ and $Y$ be independent discrete random variables with:


 * $X \sim \Poisson {\lambda_1}$

and
 * $Y \sim \Poisson {\lambda_2}$

for some $\lambda_1, \lambda_2 \in \R_{> 0}$.

Then their sum $X + Y$ is distributed:


 * $X + Y \sim \Poisson {\lambda_1 + \lambda_2}$