Probability Generating Function of Shifted Geometric Distribution

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

Then the p.g.f. of $X$ is:
 * $\Pi_X \left({s}\right) = \dfrac {ps} {1 - qs}$

where $q = 1 - p$.

Proof
From the definition of p.g.f:


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

From the definition of the shifted geometric distribution:
 * $\forall k \in \N, k \ge 1: p_X \left({k}\right) = p q^{k-1}$

So:

Hence the result.