Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.7 GENERATING FUNCTIONS

probability that value ofSNlies in the intervalstos+dsis given by§

Pr(s<SN≤s+ds)=

∑∞

n=0

Pr(N=n)Pr(s<X 0 +X 1 +X 2 ···+Xn≤s+ds).

Write Pr(s<SN≤s+ds)asfN(s)dsand Pr(s<X 0 +X 1 +X 2 ···+Xn≤s+ds)

asfn(s)ds.Thekth moment of the PDFfN(s) is given by

μk=

∫

skfN(s)ds=

∫

sk

∑∞

n=0

Pr(N=n)fn(s)ds

=

∑∞

n=0

Pr(N=n)

∫

skfn(s)ds

=

∑∞

n=0

hn×(k!×coefficient oftkin [MX(t)]n)

Thus the MGF ofSNis given by

MSN(t)=

∑∞

k=0

μk

k!

tk=

∑∞

n=0

hn

∑∞

k=0

tk×coefficient oftkin [MX(t)]n

=

∑∞

n=0

hn[MX(t)]n

=χN(MX(t)).

In words, the MGF of the sumSNis given by the compound functionχN(MX(t))

obtained by substitutingMX(t)fortin the PGF for the number of termsNin

the sum.

Uniqueness

If the MGF of the random variableX 1 is identical to that forX 2 then the

probability distributions ofX 1 andX 2 are identical. This is intuitively reasonable

although a rigorous proof is complicated,¶and beyond the scope of this book.

30.7.3 Characteristic function

Thecharacteristic function(CF) of a random variableXis defined as

CX(t)=E

[

eitX

]

=

{∑

je

itxjf(xj) for a discrete distribution,

∫

eitxf(x)dx for a continuous distribution (30.91)

§As in the previous section,X 0 has to be formally included, since Pr(N= 0) may be non-zero.

¶See, for example, P. A. Moran,An Introduction to Probability Theory(New York: Oxford Science

Publications, 1984).