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

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

So:

$$ $$ $$

Hence the result.