Page 464 - A First Course In Stochastic Models

E. LAPLACE TRANSFORM THEORY 459

as a function of the complex variable s with Re(s) > B. The integral always
exists when Re(s) > B. If f (x) is the probability density of a random variable
X, the Laplace transform f (s) is defined for all s with Re(s) > 0 and can be
∗
interpreted as
∗
f (s) = E(e −sX ). (E.1)
Moreover, we then have

k
d f (s)
∗
k
k
E(X ) = (−1) lim , k = 1, 2, . . . . (E.2)
s→0 ds k
The results (a) to (c) below can easily be verified from the definition of Laplace
transform. In the statements it is assumed that the various integrals exist.
(a) If the function f (x) = ag(x) + bh(x) is a linear combination of the functions
g(x) and h(x) with Laplace transforms g (s) and h (s), then
∗
∗
∗
f (s) = ag (s) + bh (s). (E.3)
∗
∗
x
(b) If F(x) = f (y) dy, then
0
∗

∞ f (s)
e −sx F(x) dx = . (E.4)
0 s
If f (x) has a continuous derivative f (x) then
′

∞
∗
′
e −sx f (x) dx = sf (s) − f (0). (E.5)
0
(c) If the function f (x) is given by the convolution
x

f (x) = g(x − y)h(y) dy
0
∗
of two functions g(x) and h(x) with Laplace transforms g (s) and h (s), then
∗
∗
∗
∗
f (s) = g (s)h (s). (E.6)
In addition to these results, we mention without proof the following useful
∞ −sx
Abelian theorem. If e f (x) dx is convergent for Re(s) > 0 and lim x→∞ f (x)
0
exists, then
∞

lim f (x) = lim s e −sx f (x) dx. (E.7)
x→∞ s→0 0
In applied probability problems one often encounters the situation of a non-negative
random variable X that has a positive mass at x = 0 and a density on (0, ∞). Then
−sX
∞ 1 − E(e )
e −sx P {X > x} dx = . (E.8)
0 s