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:

$\Pi_X \left({s}\right) = \dfrac q {1 - ps}$

where $q = 1 - p$.

Proof

From the definition of p.g.f:

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

From the definition of the geometric distribution:

$\forall k \in \N, k \ge 0: p_X \left({k}\right) = q p^k$

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Pi_X \left({s}\right)\)	\(=\)	\(\displaystyle \sum_{k \ge 0} q p^k s^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle q \sum_{k \ge 0} \left({ps}\right)^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle q \frac 1 {1 - ps}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by Sum of Infinite Geometric Progression

Hence the result.

$\blacksquare$