[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]S_N = X_1 + X_2 +\cdots+ X_N[/math], where the [math]X_i[/math]'s are independent random

variables with common distribution having generating function [math]f(z)[/math]. Assume that [math]N[/math] is an integer valued random variable independent of all of the [math]X_j[/math] and having generating function [math]g(z)[/math]. Show that the generating function for [math]S_N[/math] is [math]h(z) = g(f(z))[/math]. Hint: Use the fact that

[[math]] h(z) = E(z^{S_N}) = \sum_k E(z^{S_N} | N = k) P(N = k)\ . [[/math]]