Page 115 - Elements of Distribution Theory

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052184472Xc04 CUNY148/Severini May 24, 2005 2:39

4.3 Laplace Transforms and Moment-Generating Functions 101

Let X denote a real-valued random variable with Laplace transform L. The Laplace
transform has the property that moments of X may be obtained from the derivatives of L(t)

at t = 0. Note that since L(t)is defined only for t ≥ 0, L (0), L (0), and so on will refer to

the right-hand derivatives of L(t)at t = 0; for example,
L(h) − L(0)

L (0) = lim .
h→0 + h
Theorem 4.6. Let X denote a real-valued, nonnegative random variable and let L denote
m (m)
its Laplace transform. Suppose that, for some m = 1, 2,..., E[X ] < ∞. Then L (0)
exists and
m m (m)
E[X ] = (−1) L (0).
(m) m
Conversely, if L (0) exists, then E(X ) < ∞.
Proof. We will consider only the case m = 1; the general case follows along similar lines.
Note that, by the mean-value theorem, for all h, x, there exists a q ≡ q(x, h), 0 ≤ q ≤ h,
such that

exp(−hx) − 1 =−x exp(−q(x, h)x)h. (4.1)
Hence,

lim q(x, h) = 0
h→0
and, for all h > 0 and all x,
exp{−q(x, h)h}≤ 1.

By (4.1), the existence of L (0) is related to the existence of

X

∞
lim −x exp{−q(x, h)x} dF(x).
h→0 + 0
Suppose that L (0) exists. Then the limits

∞ exp{−hx}− 1 ∞
lim dF(x) = lim −x exp{−q(x, h)x} dF(x)
h→0 + 0 h h→0 + 0
exist and are finite. By Fatou’s Lemma (see Appendix 1),
∞ ∞

lim x exp{−q(x, h)x} dF(x) ≥ x lim inf exp{−q(x, h)x} dF(x)
h→0 + 0 0 h→0 +
∞

= xdF(x),
0
so that E(X) < ∞.
Suppose E(X) < ∞. Since, for h ≥ 0, x exp(−hx) ≤ x and, for all x ≥ 0,
x exp(−hx) → x as h → 0
by the Dominated Convergence Theorem (see Appendix 1),

∞
lim −x exp{−hx} dF(x) =−E(X).
h→0 +
0
Hence, L (0) exists and is equal to −E(X).

X