Fundamentals of Probability and Statistics for Engineers

of Y if they exist. However, this procedure – the determination of moments of Y
on finding the probability law of Y – is cumbersome and unnecessary if only the
moments of Y are of interest.
A more expedient and direct way of finding the moments of Y g(X), given
the probability law of X, is to express moments of Y as expectations of
appropriate functions of X; they can then be evaluated directly within the
probability domain of X. In fact, all the ‘machinery’ for proceeding along this
line is contained in Equations (4.1) and (4.2).
Let Y g(X) and assume that all desired moments of Y exist. The nth
moment of Y can be expressed as

It fo llows from Equations (4.1) and (4.2) that, in terms of the pmf or pdf of X,

An alternative approach is to determine the characteristic function of Y from
which all moments of Y can be generated through differentiation. As we see
from the definition [Equations (4.46) and (4.47)], the characteristic function of
Y can be expressed by

1

3

4

1

4

ˇ 2

FV 2 (ˇˇ 2 )

1

Figure 5. 15 Distribution FV 2 (v 2 ) in Example 5.8

Functions of Random Variables 135

±

±

ˆ

ˆ

EfYngˆEfgn…X†g:… 5 : 30 †

EfYngˆEfgn…X†gˆ

X

i

gn…xi†pX…xi†; Xdiscrete;

EfYngˆEfgn…X†gˆ

Z 1

1

gn…x†fX…x†dx; Xcontinuous:

… 5 : 31 †

Y…t†ˆEfejtYgˆEfejtg…X†gˆ

X

i

ejtg…xi†pX…xi†; Xdiscrete;

Y…t†ˆEfejtYgˆEfejtg…X†gˆ

Z 1

1

ejtg…x†fX…x†dx; Xcontinuous:

9

>>=

>>

;

… 5 : 32 †

ν

V ν