Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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)=

∏n

i=1

ΦXi(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 variables

As 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=0

Pr(N=n)Pr(X 0 +X 1 +X 2 +···+Xn=k)

=

∑∞

n=0

hn×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.