Page 119 - Fundamentals of Probability and Statistics for Engineers

102 Fundamentals of Probability and Statistics for Engineers

Fourier transforms that f (x) is uniquely determined from Equation (4.58);
X
that is, no two distinct density functions can have the same characteristic
function.
This property of the characteristic function provides us with an alternative
way of arriving at the distribution of a random variable. In many physical
problems, it is often more convenient to determine the density function of a
random variable by first determining its characteristic function and then per-
forming the Fourier transform as indicated by Equation (4.58). Furthermore,
we shall see that the characteristic function has properties that render it
particularly useful for determining the distribution of a sum of independent
random variables.
The inversion formula of Equation (4.58) follows immediately from the
theory of Fourier transforms, but it is of interest to give a derivation of this
equation from a probabilistic point of view.
Proof of Equation (4.58): an integration formula that can be found in any
table of integrals is
8 1; for a < 0;
1 Z 1 sin at <
dt ˆ 0; for a ˆ 0; 4:59†
1 t : 1; for a > 0:

This leads to
8 1; for a < 0;
1 Z 1 sin at ‡ j 1 cos at† <
dt ˆ 0; for a ˆ 0; 4:60†
1 t : 1; for a > 0;

)/

because the function (1 cos at t is an odd function of t so that its integral

over a symmetric range vanishes. Upon replacing a by X x in Equation
(4.60), we have
8
1; for X < x;
>
1
1 j Z 1 e j X x†t < 1
>
dt ˆ ; for X ˆ x; 4:61†
2 2 1 t > 2
>
0; for X > x:
:
For a fixed value of x, Equation (4.61) is a function of random variable X, and
it may be regarded as defining a new random variable Y . The random variable
1
Y is seen to be discrete, taking on values 1, , and 0 with probabilities
2
P(X < x), P(X ˆ x), and P(X > x), respectively. The mean of Y is thus equal to

1
EfYgˆ 1†P X < x†‡ P X ˆ x†‡ 0†P X > x†:
2

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