30.7 GENERATING FUNCTIONS
i.e. the PGF of the sum of two independent random variables is equal to the
product of their individual PGFs. The same result can be deduced in a less formal
way by noting that ifXandYare independent then
E[
tX+Y]
=E[
tX]
E[
tY]
.Clearly result (30.80) can be extended to more than two random variables by
writingS 3 =S 2 +Zetc., to give
Φ(∑ni=1Xi)(t)=∏ni=1ΦXi(t), (30.81)and, further, if all theXihave the same probability distribution,
Φ(∑ni=1Xi)(t)=[ΦX(t)]n. (30.82)This latter result has immediate application in the deduction of the PGF for the
binomial distribution from that for a single trial, equation (30.72).
Variable-length sums of random variablesAs a final result in the theory of probability generating functions we show how to
calculate the PGF for a sum ofNrandom variables, all with the same probability
distribution, when the value ofNis itself a random variable but one with a
known probability distribution. In symbols, we wish to find the distribution
of
SN=X 1 +X 2 +···+XN, (30.83)whereN is a random variable with Pr(N =n)=hn and PGFχN(t)=
∑
hntn.
The probabilityξkthatSN=kis given by a sum of conditional probabilities,
namely§
ξk=∑∞n=0Pr(N=n)Pr(X 0 +X 1 +X 2 +···+Xn=k)=∑∞n=0hn×coefficient oftkin [ΦX(t)]n.Multiplying both sides of this equation bytkand summing over allk,weobtain
§FormallyX 0 = 0 has to be included, since Pr(N= 0) may be non-zero.