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:
 * $\map {\Pi_X} s = \paren {q + p s}^n$

where $q = 1 - p$.

Proof
From the definition of p.g.f:
 * $\ds \map {\Pi_X} s = \sum_{k \mathop \ge 0} \map {p_X} k s^k$

From the definition of the binomial distribution:
 * $\map {p_X} k = \dbinom n k p^k \paren {1 - p}^{n - k}$

So:

Hence the result.