Page 113 - Elements of Distribution Theory

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4.3 Laplace Transforms and Moment-Generating Functions 99

of (X i , X j ). Let

σ ij = Cov(X i , X j ), i, j = 1,..., d.

It is often convenient to work with these covariances in matrix form. Hence, let denote
the d × d matrix with (i, j)th element given by σ ij ; note that this matrix is symmetric. The
matrix is known as the covariance matrix of X. The following theorem gives some basic
properties of covariance matrices; the proof is left as an exercise.

T
Theorem 4.4. Let X denote a d-dimensional random vector such that E(X X) < ∞ and
let denote the covariance matrix of X. Then
T
T
(i) Var(a X) = a a, a ∈ R d
(ii) is nonnegative definite

4.3 Laplace Transforms and Moment-Generating Functions
Let X denote a real-valued random variable. Consider the expected value E[exp{tX}] where
t is a given number, −∞ < t < ∞. Since exp{tX} is a nonnegative random variable, this
expected value always exists, although it may be infinite.
The expected value of exp{tX} is closely related to the characteristic function of X,
the expected value of exp{it X}.However, there is an important difference between the
functions exp{tx} and exp{itx}. Although exp{itx} is bounded, with | exp{itx}| = 1, the
function exp{tx} is unbounded for any nonzero t and grows very fast as either x →∞ or
x →−∞. Hence, for many random variables, the set of values of t for which E{exp(tX)}
is finite is quite small.
Suppose there exists number δ> 0 such that E[exp{tX}] < ∞ for |t| <δ.In this case,
we say that X,or, more precisely, the distribution of X, has moment-generating function

M X (t) = E[exp{tX}], |t| <δ.

As noted above, it is not unusual for a random variable to not have a moment-generating
function.

Laplace transforms
The situation is a little better if X is nonnegative. In that case, we know that E{exp(tX)} < ∞
for all t ≤ 0. Hence, we may define a function
L(t) = E{exp(−tX)}, t ≥ 0;

we will refer to this function as the Laplace transform of the distribution or, more simply,
the Laplace transform of X.

Example 4.6 (Gamma distribution). Consider the gamma distribution with parameters α
and β,as discussed in Example 3.4. The Laplace transform of this distribution is given by
∞ β β

α α
L(t) = exp(−tx) x α−1 exp(−βx) dx = α , t ≥ 0.
0 (α) (β + t)