Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

n

r=n

r

Figure 30.10 The pairs of values ofnandrused in the evaluation of ΦX+Y(t).

Sums of random variables

We now turn to considering the sum of two or more independent random

variables, sayXandY, and denote byS 2 the random variable

S 2 =X+Y.

If ΦS 2 (t)isthePGFforS 2 , the coefficient oftnin its expansion is given by the

probability thatX+Y=nand is thus equal to the sum of the probabilities that

X=randY=n−rfor all values ofrin 0≤r≤n. Since such outcomes for

different values ofrare mutually exclusive, we have

Pr(X+Y=n)=

∑∞

r=0

Pr(X=r)Pr(Y=n−r). (30.79)

Multiplying both sides of (30.79) bytnand summing over all values ofnenables

us to express this relationship in terms of probability generating functions as

follows:

ΦX+Y(t)=

∑∞

n=0

Pr(X+Y=n)tn=

∑∞

n=0

∑n

r=0

Pr(X=r)trPr(Y=n−r)tn−r

=

∑∞

r=0

∑∞

n=r

Pr(X=r)trPr(Y=n−r)tn−r.

The change in summation order is justified by reference to figure 30.10, which

illustrates that the summations are over exactly the same pairs of values ofnand

r, but with the first (inner) summation over the points in a column rather than

over the points in a row. Now, settingn=r+sgives the final result,

ΦX+Y(t)=

∑∞

r=0

Pr(X=r)tr

∑∞

s=0

Pr(Y=s)ts

=ΦX(t)ΦY(t), (30.80)