Page 465 - A First Course In Stochastic Models

460 APPENDICES

e
Using that E(e −sX ) = P {X = 0}+ 0 ∞ −sx π(x) dx and P {X > x} = 1−P {X =
x
0} − π(y) dy with π(x) denoting the derivative of P {X ≤ x} for x > 0, the
0
relation (E.8) follows directly from (E.3) and (E.4). Of course the result (E.8) also
holds when X has a zero mass at x = 0.
In specific applications requiring the determination of some unknown function
f (x) it is often possible to obtain the Laplace transform f (s) of f (x). A very use-
∗
ful result is that a continuous function f (x) is uniquely determined by its Laplace
transform f (s). In principle the function f (x) can be obtained by inversion of its
∗
Laplace transform. Extensive tables are available for the inverse of basic forms of
f (s); see for example Abramowitz and Stegun (1965). An inversion formula that
∗
is sometimes helpful in applications is the Heaviside formula. Suppose that
P (s)
∗
f (s) = ,
Q(s)
where P (s) and Q(s) are polynomials in s such that the degree of P (s) is smaller
than that of Q(s). It is no restriction to assume that P (s) and Q(s) have no zeros
in common. Let s 1 , . . . , s k be the distinct zeros of Q(s) in the complex plane. For
ease of presentation, assume that each root s j is simple (i.e. has multiplicity 1).
Then it is known from algebra that P (s)/Q(s) admits the partial fraction expansion
P (s) r 1 r 2 r k
= + + · · · + ,
Q(s) s − s 1 s − s 2 s − s k
(s − s j )P (s)/Q(s) and so r j = P (s j )/Q (s j ), 1 ≤ j ≤ k. The
′
where r j = lim s→s j
inverse of the Laplace transform f (s) = P (s)/Q(s) is now given by
∗
k
P (s j )
s j x
f (x) = e , (E.9)
Q (s j )
′
j=1
as can be verified by taking the Laplace transform of both sides of this equation.
This result is readily extended to the case in which some of the roots of Q(s) = 0
are not simple. For example, the inverse of the Laplace transform
P (s)
∗
f (s) = ,
(s − a) m
where P (s) is a polynomial of degree lower than m, is given by
m (m−j) j−1
P (a)x
ax
f (x) = e . (E.10)
(m − j)!(j − 1)!
j=1
Here P (n) (a) denotes the nth derivative of P (x) at x = a with P (0) (a) = P (a).
It is usually not possible to give an explicit expression for the inverse of a
given Laplace transform. In those situations the unknown function f (x) may be