PGF of Sum of Random Number of Independent Discrete Random Variables

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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:

$\map {\Pi_Z} s = \map {\Pi_N} {\map {\Pi_X} s}$


Proof

\(\ds \map {\Pi_Z} s\) \(=\) \(\ds \expect {s^{X_1 + X_2 + \cdots + X_N} }\) Definition of Probability Generating Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} \expect {s^{X_1 + X_2 + \cdots + X_N} \mid N = n} \map \Pr {N = n}\) Total Expectation Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} \expect {s^{X_1 + X_2 + \cdots + X_n} } \map \Pr {N = n}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} \map {\Pi_X} s^n \map \Pr {N = n}\) PGF of Sum of Independent Discrete Random Variables
\(\ds \) \(=\) \(\ds \map {\Pi_N} {\map {\Pi_X} s}\) Definition of $\map {\Pi_N} s$

$\blacksquare$


Sources

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