Probability Generating Function of Geometric Distribution

Theorem
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$.

Then the p.g.f. of $X$ is:
 * $\map {\Pi_X} s = \dfrac q {1 - p s}$

where $q = 1 - p$.

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

From the definition of the geometric distribution:
 * $\forall k \in \N, k \ge 0: \map {p_X} k = q p^k$

So:

Hence the result.