Page 152 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 135

ˇˇ ν
F ( )
V V 2
2
1
3
4

1
4
ˇ ν 2
1

(v 2 ) in Example 5.8
Figure 5.15 Distribution F V 2
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 ˆ 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 ˆ g(X ) and assume that all desired moments of Y exist. The nth
moment of Y can be expressed as

n
n
EfY g ˆ Efg X†g: 5:30†
It follows from Equations (4.1) and (4.2) that, in terms of the pmf or pdf of X,

X
n
n
n
EfY gˆ Efg X†g ˆ g x i †p x i †; X discrete;
X
i
1 5:31†
Z
n
n
n
EfY gˆ Efg X†g ˆ g x†f x†dx; X continuous:
X

1
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
Y t†ˆ Efe jtY gˆ Efe jtg X† gˆ X e jtg x i † p x i †; X discrete; 9
X
>
>
=
i
Z 1 5:32†
Y t†ˆ Efe jtY gˆ Efe jtg X† gˆ e jtg x† f x†dx; >
>
X X continuous: ;

1
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