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

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:

Hence the result.