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 $Z$.

Generalization
Let:
 * $Z = X_1 + X_2 + \ldots + X_n$

where each of $X_1, X_2, \ldots, X_n$ are independent discrete random variables with PGFs $\Pi_{X_1} \left({s}\right), \Pi_{X_2} \left({s}\right), \ldots, \Pi_{X_n} \left({s}\right)$.

Then:
 * $\displaystyle \Pi_Z \left({s}\right) = \prod_{k=1}^n \Pi_{X_k} \left({s}\right)$

Proof of Generalization
Straightforward inductive proof.