Page 121 - Fundamentals of Probability and Statistics for Engineers

104 Fundamentals of Probability and Statistics for Engineers

and therefore
1 Z u
p x†ˆ lim Efe j X x†t gdt
X
u!1 2u
u
4:67†
1 Z u jtx
ˆ lim e X t†dt;
u!1 2u
u
which gives the desired inversion formula.
In summary, the transform pairs given by Equations (4.46), (4.47), (4.58),
and (4.64) are collected and presented below for easy reference. For a contin-
uous random variable X,

Z 1 9
jtx
X t†ˆ e f x†dx; >
X
>
1 >
=
4:68†
1 Z 1 jtx >
f x†ˆ e >
>
X X t†dt; ;
2 1
and, for a discrete random variable X,
X
9
X t†ˆ e jtx i p x i †;
X >
>
i = 4:69†
1 Z u jtx >
p x†ˆ lim e X t†dt: ;
>
X
u!1 2u
u
Of the two sets, Equations (4.68) for the continuous case are more important in
terms of applicability. As we shall see in Chapter 5, probability mass functions
for discrete random variables can be found directly without resorting to their
characteristic functions.
As we have mentioned before, the characteristic function is particularly
useful for the study of a sum of independent random variables. In this connec-
tion, let us state the following important theorem, (Theorem 4.3).
Theorem 4.3: The characteristic function of a sum of independent random
variables is equal to the product of the characteristic functions of the individual
random variables.

Proof of Theorem 4.3: Let

Y ˆ X 1 ‡ X 2 ‡ ‡ X n : 4:70†

TLFeBOOK