Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

an expression for the PGF ΞS(t)ofSN:

ΞS(t)=

∑∞

k=0

ξktk=

∑∞

k=0

tk

∑∞

n=0

hn×coefficient oftkin [ΦX(t)]n

=

∑∞

n=0

hn

∑∞

k=0

tk×coefficient oftkin [ΦX(t)]n

=

∑∞

n=0

hn[ΦX(t)]n

=χN(ΦX(t)). (30.84)

In words, the PGF of the sumSNis given by the compound functionχN(ΦX(t))

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

the sum. We illustrate this with the following example.

The probability distribution for the number of eggs in a clutch is Poisson distributed with

meanλ, and the probability that each egg will hatch isp(and is independent of the size of

the clutch). Use the results stated in (30.72) and (30.73) to show that the PGF (and hence

the probability distribution) for the number of chicks that hatch corresponds to a Poisson

distribution having meanλp.

The number of chicks that hatch is given by a sum of the form (30.83) in whichXi=1if
theith chick hatches andXi= 0 if it does not. As given by (30.72), ΦX(t) is thus (1−p)+pt.
The value ofNis given by a Poisson distribution with meanλ; thus, from (30.73), in the
terminology of our previous discussion,

χN(t)=e−λeλt.

We now substitute these forms into (30.84) to obtain

ΞS(t)=exp(−λ)exp[λΦX(t)]

=exp(−λ)exp{λ[(1−p)+pt]}

=exp(−λp)exp(λpt).

But this is exactly the PGF of a Poisson distribution with meanλp.
That this implies that the probability is Poisson distributed is intuitively obvious since,
in the expansion of the PGF as a power series int, every coefficient will be precisely
that implied by such a distribution. A solution of the same problem by direct calculation
appears in the answer to exercise 30.29.

30.7.2 Moment generating functions

As we saw in section 30.5 a probability function is often expressed in terms of

its moments. This leads naturally to the second type of generating function, a

moment generating function. For a random variableX, and a real numbert,the

moment generating function (MGF) is defined by

MX(t)=E

[

etX

]

=

{∑

ie

txif(x

i) for a discrete distribution,

∫

etxf(x)dx for a continuous distribution. (30.85)