Page 115 - Fundamentals of Probability and Statistics for Engineers

98 Fundamentals of Probability and Statistics for Engineers

exhibits a random variation measured by v, then a physical process resulting
from additive actions of n molecules will possess a random variation measured
1/2
by v/n . It decreases as n increases. Since n is generally very large in the
workings of physical processes, this result leads to the conjecture that the laws
of physics can be exact laws despite local disorder.

4.5 CHARACTERISTIC FUNCTIONS

The expectation Efe jtX g of a random variable X is defined as the characteristic
function of X. Denoted by X (t), it is given by

X t†ˆ Efe jtX gˆ X e jtx i p x i †; X discrete; 4:46†
X
i
Z 1
jtx
X t†ˆ Efe jtX gˆ e f x†dx; X continuous; 4:47†
X
1
p 
where t is an arbitrary real-valued parameter and j ˆ 1. The characteristic
function is thus the expectation of a complex function and is generally complex
valued. Since

je jtX jˆj cos tX ‡ j sin tXjˆ 1;

the sum and the integral in Equations (4.46) and (4.47) exist and therefore X (t)
always exists. Furthermore, we note

X 0†ˆ 1; 9
>
=

X t†ˆ t†; 4:48†
X
>
j X t†j 1;
;
where the asterisk denotes the complex conjugate. The first two properties are
self-evident. The third relation follows from the observation that, since
f (x) 0,
X
1 Z 1
Z
jtx
j X t†j ˆ e f x†dx f x†dx ˆ 1:

X
X
1 1

The proof is the same as that for discrete random variables.
We single this expectation out for discussion because it possesses a number of
important properties that make it a powerful tool in random-variable analysis
and probabilistic modeling.

TLFeBOOK