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:

$\ds \map {\Pi_X} s = \sum_{x \mathop \ge 0} \map {p_X} x s^x$

From the definition of the geometric distribution:

$\forall k \in \N, k \ge 0: \map {p_X} k = q p^k$

So:

\(\ds \map {\Pi_X} s\)

\(=\)

\(\ds \sum_{k \mathop \ge 0} q p^k s^k\)

\(\ds \)

\(=\)

\(\ds q \sum_{k \mathop \ge 0} \paren {p s}^k\)

\(\ds \)

\(=\)

\(\ds q \frac 1 {1 - p s}\)

Sum of Infinite Geometric Sequence

Hence the result.

$\blacksquare$