I found this in one of the review exercises.

Let X be a discrete random variable with values in $\displaystyle \mathbb Z_{+} := {0,1,2, ...}$ with probability generating function

$\displaystyle P_{X}(z) = \mathbb E(z^{X}) = \sum^{+ \infty}_{k=0} \mathbb P (X = k)z^{k}$

(i) Show that $\displaystyle P_X$ defines a mapping from [0,1] to [0,1].

(ii) Let $\displaystyle (X_n)$ be a sequence of i.i.d random variables with values in $\displaystyle \mathbb Z_{+}$ and $\displaystyle T$ a random variable with values in $\displaystyle \mathbb Z_{+}$ which is independent of $\displaystyle (X_n)$. Define

$\displaystyle S_{0} = 0$ and $\displaystyle S_{T} = \sum^{T}_{j=1} X_{j}$

Show that if we denote by $\displaystyle P_{T}$ and $\displaystyle P_{X}$ the probability the generating functions of $\displaystyle T$ and $\displaystyle X_{1}$ respectively, then the probability generating function of S is given by

$\displaystyle \forall z \in [0,1], P_{S}(z) = P_{T}(P_{X}(z))$.

Thank you (: