# Probability Generating Function of Binomial Distribution

## Theorem

Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then the p.g.f. of $X$ is:

$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$

where $q = 1 - p$.

## Proof

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{k \mathop \ge 0} p_X \left({k}\right) s^k$

From the definition of the binomial distribution:

$\displaystyle p_X \left({k}\right) = \binom n k p^k \left({1 - p}\right)^{n-k}$

So:

 $\displaystyle \Pi_X \left({s}\right)$ $=$ $\displaystyle \sum_{k \mathop = 0}^n \binom n k p^k \left({1 - p}\right)^{n - k} s^k$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop = 0}^n \binom n k \left({p s}\right)^k \left({1 - p}\right)^{n - k}$ $\displaystyle$ $=$ $\displaystyle \left({\left({p s}\right) + \left({1 - p}\right)}\right)^n$ by the Binomial Theorem

Hence the result.

$\blacksquare$