Sum of Independent Binomial Random Variables

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


 * $X \sim \Binomial m p$

and
 * $Y \sim \Binomial n p$

Let $X$ and $Y$ be independent.

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


 * $Z \sim \Binomial {m + n} p$

Proof
From Probability Generating Function of Poisson Distribution, we have that the probability generating functions of $X$ and $Y$ are given by:
 * $\map {\Pi_X} s = \paren {q + p s}^m$
 * $\map {\Pi_Y} s = \paren {q + p s}^n$

respectively.

Now because of their independence, we have:

This is the probability generating function for a discrete random variable with a binomial distribution:
 * $\Binomial {m + n} p$

Therefore:
 * $Z = X + Y \sim \Binomial {m + n} p$