Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

If, as previously, the probability that the random variableXtakes the valuexn

isf(xn), then
∑

n

f(xn)=1.

In the present case, however, only non-negative integer values ofxnare possible,

and we can, without ambiguity, write the probability thatXtakes the valuenas

fn, with

∑∞

n=0

fn=1. (30.70)

We may now define theprobability generating functionΦX(t)by

ΦX(t)≡

∑∞

n=0

fntn. (30.71)

It is immediately apparent that ΦX(t)=E[tX] and that, by virtue of (30.70),

ΦX(1) = 1.

Probably the simplest example of a probability generating function (PGF) is

provided by the random variableXdefined by

X=

{

1 if the outcome of a single trial is a ‘success’,

0 if the trial ends in ‘failure’.

If the probability of success ispand that of failureq(= 1−p)then

ΦX(t)=qt^0 +pt^1 +0+0+···=q+pt. (30.72)

This type of random variable is discussed much more fully in subsection 30.8.1.

In a similar but slightly more complicated way, a Poisson-distributed integer

variable with meanλ(see subsection 30.8.4) has a PGF

ΦX(t)=

∑∞

n=0

e−λλn

n!

tn=e−λeλt. (30.73)

We note that, as required, ΦX(1) = 1 in both cases.

Useful results will be obtained from this kind of approach only if the summation

(30.71) can be carried out explicitly in particular cases and the functions derived

from ΦX(t) can be shown to be related to meaningful parameters. Two such

relationships can be obtained by differentiating (30.71) with respect tot. Taking

the first derivative we find

dΦX(t)

dt

=

∑∞

n=0

nfntn−^1 ⇒ Φ′X(1) =

∑∞

n=0

nfn=E[X], (30.74)