Sum of Independent Poisson Random Variables is Poisson

Theorem
Let $X$ and $Y$ be discrete random variables with a Poisson distribution:


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

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

Let $X$ and $Y$ be independent.

Then their sum $Z = X + Y$ is distributed as:


 * $Z \sim \Poisson {\lambda_1 + \lambda_2}$

Proof
From Probability Generating Function of Poisson Distribution, we have that the probability generating functions of $X$ and $Y$ are given by:
 * $\map {\Pi_X} s = e^{-\lambda_1 \paren {1 - s} }$
 * $\map {\Pi_Y} s = e^{-\lambda_2 \paren {1 - s} }$

respectively.

Now because of their independence, we have:

This is the probability generating function for a discrete random variable with a Poisson distribution:
 * $\Poisson {\lambda_1 + \lambda_2}$

Therefore:
 * $Z = X + Y \sim \Poisson {\lambda_1 + \lambda_2}$