PGF of Sum of Independent Discrete Random Variables

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ and $Y$ be independent discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.

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

Then:
 * $\map {\Pi_Z} s = \map {\Pi_X} s \, \map {\Pi_Y} s$

where $\map {\Pi_Z} s$ is the probability generating function of $Z$.