Grinstead-Snell-Probability · Problem 10.2.4

Generating Functions

Mathematics · branching-processes

Problem

Let

S_N = X_1 + X_2 + \cdots + X_N

, where the

X_i

's are independent random variables with common distribution having generating function

f(z)

. Assume that

N

is an integer valued random variable independent of all of the

X_j

and having generating function

g(z)

. Show that the generating function for

S_N

h(z) = g(f(z))

. Hint: Use the fact that

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