I found this in one of the review exercises.

Let X be a discrete random variable with values in with probability generating function

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

(ii) Let be a sequence of i.i.d random variables with values in and a random variable with values in which is independent of . Define

and

Show that if we denote by and the probability the generating functions of and respectively, then the probability generating function of S is given by

.

Thank you (: