Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.7 GENERATING FUNCTIONS

The MGF will exist for all values oftprovided thatXis bounded and always

exists at the pointt=0whereM(0) =E(1) = 1.

It will be apparent that the PGF and the MGF for a random variableX

are closely related. The former is the expectation oftXwhilst the latter is the

expectation ofetX:

ΦX(t)=E

[

tX

]

,MX(t)=E

[

etX

]

.

The MGF can thus be obtained from the PGF by replacingtbyet, and vice

versa. The MGF has more general applicability, however, since it can be used

with both continuous and discrete distributions whilst the PGF is restricted to

non-negative integer distributions.

As its name suggests, the MGF is particularly useful for obtaining the moments

of a distribution, as is easily seen by noting that

E

[

etX

]

=E

[

1+tX+

t^2 X^2

2!

+···

]

=1+E[X]t+E

[

X^2

]t^2

2!

+···.

Assuming that the MGF exists for alltaround the pointt= 0, we can deduce

that the moments of a distribution are given in terms of its MGF by

E[Xn]=

dnMX(t)

dtn

∣

∣

∣

∣

t=0

. (30.86)

Similarly, by substitution in (30.51), the variance of the distribution is given by

V[X]=M′′X(0)−

[

M′X(0)

] 2

, (30.87)

where the prime denotes differentiation with respect tot.

The MGF for the Gaussian distribution (see the end of subsection 30.9.1) is given by

MX(t)=exp

(

μt+^12 σ^2 t^2

)

.

Find the expectation and variance of this distribution.

Using (30.86),

MX′(t)=

(

μ+σ^2 t

)

exp

(

μt+^12 σ^2 t^2

)

⇒ E[X]=M′X(0) =μ,

MX′′(t)=

[

σ^2 +(μ+σ^2 t)^2

]

exp

(

μt+^12 σ^2 t^2

)

⇒ MX′′(0) =σ^2 +μ^2.

Thus, using (30.87),

V[X]=σ^2 +μ^2 −μ^2 =σ^2.

That the mean is found to beμand the varianceσ^2 justifies the use of these symbols in
the Gaussian distribution.

The moment generating function has several useful properties that follow from

its definition and can be employed in simplifying calculations.