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:
 * $$\Pi_X \left({s}\right) = \frac {ps} {1 - qs}$$

where $$q = 1 - p$$.

Proof
From the definition of p.g.f:


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

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.