PGF of Sum of Random Number of Independent Discrete Random Variables

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

Let:
 * $N, X_1, X_2, \ldots$

be independent discrete random variables such that the $X$'s have the same probability distribution.

Let:
 * $\map {\Pi_N} s$ be the PGF of $N$
 * $\map {\Pi_X} s$ be the PGF of each of the $X$'s.

Let:
 * $Z = X_1 + X_2 + \ldots + X_N$

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