Page 117 - Elements of Distribution Theory

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4.3 Laplace Transforms and Moment-Generating Functions 103

Moment-generating functions
The main drawback of Laplace transforms is that they apply only to nonnegative random
variables. The same idea used to define the Laplace transform can be applied to a random
variable with range R, yielding the moment-generating function; however, as noted earlier,
moment-generating functions do not always exist.
Let X denote a real-valued random variable and suppose there exists a number δ> 0
such that E[exp{tX}] < ∞ for |t| <δ.In this case, we say that X has moment-generating
function
M(t) ≡ M X (t) = E[exp{tX}], |t| <δ;
δ is known as the radius of convergence of M X . The moment-generating function is closely
related to the characteristic function of X and, if X is nonnegative, to the Laplace transform
of X.

Example 4.10 (Poisson distribution). Let X denote a discrete random variable taking
values in the set {0, 1, 2,...} and let
x
p(x) = λ exp(−λ)/x!, x = 0, 1, 2,...
denote the frequency function of the distribution, where λ> 0. This is a Poisson distribution
with parameter λ.
Note that, for any value of t,
∞
x
E[exp(tX)] = exp(tx)λ exp(−λ)/x!
x=0
∞
x
= [exp(t)λ] exp(−λ)/x! = exp{[exp(t) − 1]λ}.
x=0
Hence, the moment-generating function of this distribution exists and is given by

M(t) = exp{[exp(t) − 1]λ}, −∞ < t < ∞.

Example 4.11 (Exponential distribution). Let X denote a random variable with a standard
exponential distribution. Recall that this distribution is absolutely continuous with density
function
p(x) = exp(−x) x > 0.

Note that
∞

E[exp(tX)] = exp(tx)exp(−tx) dx;
0
clearly, for any t > 1, E[exp(tX)] =∞; hence, the moment-generating function of this
distribution is given by
1
M(t) = , |t| < 1.
1 − t
This function can be compared with the Laplace transform of the distribution, which can
be obtained from Example 4.6 by taking α = β = 1:
1
L(t) = , t ≥ 0.
1 + t