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4.3 Laplace Transforms and Moment-Generating Functions 109

Example 4.15 (Poisson distribution). Let X 1 and X 2 denote independent random variables
such that, for j = 1, 2, X j has a Poisson distribution with mean λ j ; see Example 4.10. Then
X j has moment-generating function

M j (t) = exp{[exp(t) − 1]λ j }, −∞ < t < ∞.
Let X = X 1 + X 2 .By Theorem 4.11, the moment-generating function of X is given by

M(t) = M 1 (t)M 2 (t) = exp{[exp(t) − 1](λ 1 + λ 2 )}, −∞ < t < ∞.
Note that this is the moment-generating function of a Poisson distribution with mean
λ 1 + λ 2 . Thus, the sum of two independent Poisson random variables also has a Poisson
distribution.

Example 4.16 (Sample mean of normal random variables). Let Z denote a random vari-
able with standard normal distribution; then the moment-generating function of Z is given
by
1 1 2 2
∞

M(t) = exp(tz)√ exp − z dx = exp(t /2), −∞ < t < ∞.
(2π) 2
−∞
Let µ and σ denote real-valued constants, σ> 0, and let X denote a random variable
with a normal distribution with mean µ and standard deviation σ. Recall that X has the
same distribution as µ + σ Z; see Example 3.6. According to Theorem 4.10, the moment-
generating function of X is given by
2 2
M X (t) = exp(µt)exp(σ t /2), −∞ < t < ∞.
Let X 1 , X 2 ,..., X n denote independent, identically distributed random variables, each
with the same distribution as X. Then, by repeated application of Theorem 4.11, n
j=1 X j
has moment-generating function
2 2
exp(nµt)exp(nσ t /2), −∞ < t < ∞
¯ n
and, by Theorem 4.10, the sample mean X = X j /n has moment-generating function
j=1
2
2
M ¯ X (t) = exp(µt)exp[(σ /n)t /2], −∞ < t < ∞.
¯
Comparing this to M X above, we see that X has a normal distribution with mean µ and
√
standard deviation σ/ n.

Moment-generating functions for random vectors
Moment-generating functions are defined for random vectors in a manner that is analo-
gous to the definition of a characteristic function for a random vector. Let X denote a
d
d-dimensional random vector and let t denote an element of R .If there exists a δ> 0 such
that
T
E(exp{t X}) < ∞ for all ||t|| <δ,
then the moment-generating function of X exists and is given by

T
d
M X (t) = E(exp{t X}), t ∈ R , ||t|| <δ;