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

Proof
$$ $$ $$ $$ $$

Proof of Generalization
Straightforward inductive proof.