PGF of Sum of Independent Discrete Random Variables

Theorem
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $X$ and $Y$ be independent discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

Let $Z$ be a discrete random variable such that $Z = X + Y$.

Then:
 * $\Pi_Z \left({s}\right) = \Pi_X \left({s}\right) \Pi_Y \left({s}\right)$

where $\Pi_Z \left({s}\right)$ is the probability generating function of $Z$.