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:


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

From the definition of the binomial distribution:
 * $$p_X \left({k}\right) = \binom n k p^k \left({1-p}\right)^{n-k}$$

So:

$$ $$ $$

Hence the result.