PGF of Sum of Random Number of Independent Discrete Random Variables

Theorem
Let $$\left({\Omega, \Sigma, \Pr}\right)$$ 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:
 * $$\Pi_{N} \left({s}\right)$$ be the PGF of $$N$$;
 * $$\Pi_{X} \left({s}\right)$$ be the PGF of each of the $$X$$'s.

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

Then:
 * $$\Pi_Z \left({s}\right) = \Pi_{N} \left({\Pi_{X} \left({s}\right)}\right)$$

Proof
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