Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.7 GENERATING FUNCTIONS

variance of both sides of (30.66), and using (30.68), we find

V[Z(X, Y)]≈

(

∂Z

∂X

) 2

V[X]+

(

∂Z

∂Y

) 2

V[Y], (30.69)

the partial derivatives being evaluated atX=μXandY=μY.

30.7 Generating functions

As we saw in chapter 16, when dealing with particular sets of functionsfn,

each member of the set being characterised by a different non-negative integer

n, it is sometimes possible to summarise the whole set by a single function of a

dummy variable (sayt), called a generating function. The relationship between

the generating function and thenth memberfnof the set is that if the generating

function is expanded as a power series intthenfnis the coefficient oftn.For

example, in the expansion of the generating functionG(z, t)=(1− 2 zt+t^2 )−^1 /^2 ,

the coefficient oftnis thenth Legendre polynomialPn(z), i.e.

G(z, t)=(1− 2 zt+t^2 )−^1 /^2 =

∑∞

n=0

Pn(z)tn.

We found that many useful properties of, and relationships between, the members

of a set of functions could be established using the generating function and other

functions obtained from it, e.g. its derivatives.

Similar ideas can be used in the area of probability theory, and two types of

generating function can be usefully defined, one more generally applicable than

the other. The more restricted of the two, applicable only to discrete integral

distributions, is called a probability generating function; this is discussed in the

next section. The second type, a moment generating function, can be used with

both discrete and continuous distributions and is considered in subsection 30.7.2.

From the moment generating function, we may also construct the closely re-

lated characteristic and cumulant generating functions; these are discussed in

subsections 30.7.3 and 30.7.4 respectively.

30.7.1 Probability generating functions

As already indicated, probability generating functions are restricted in applicabil-

ity to integer distributions, of which the most common (the binomial, the Poisson

and the geometric) are considered in this and later subsections. In such distribu-

tions a random variable may take only non-negative integer values. The actual

possible values may be finite or infinite in number, but, for formal purposes,

all integers, 0, 1 , 2 ,...are considered possible. If only a finite number of integer

values can occur in any particular case then those that cannot occur are included

but are assigned zero probability.