Fundamentals of Probability and Statistics for Engineers

As expected, these answers agree with the results obtained earlier [Equations
(5.34) and (5.35)].
Let us again remark that the procedures described above do not require
knowledge of fY (y). One can determine fY (y) before moment calculations but it
is less expedient when only moments of Y are desired. Another remark to be
made is that, since the transformation is linear (Y 2 X 1) in this case, only
the fir st two moments of X are needed in finding the fir st two moments of Y,
that is,

as seen from Equations (5.34) and (5.35). When the transformation is nonlinear,
however, moments of X of different orders will be neede d, as shown below.

Ex ample 5. 10. Problem: from Example 5.7, determine the mean and variance
of Y X^2. The mean of Y is, in terms of fX(x),

and the second moment of Y is given by

Thus,

In this case, complete knowledge of fX (x) is not needed but we to need to
know the second and fourth moments of X.

5.2 Functions of Two or More Random Variables

In this section, we extend earlier results to a more general case. The random
variable Y is now a function of n jointly distributed random variables,
X 1 ,X 2 ,...,Xn. Formulae will be developed for the corresponding distribution
for Y.
As in the single random variable case, the case in which X 1 ,X 2 ,..., and Xn
are discrete random variables presents no problem and we will demonstrate this
by way of an example (Example 5.13). Our basic interest here lies in the

Functions of Random Variables 137

ˆ‡

EfYgˆEf 2 X‡ 1 gˆ 2 EfXg‡ 1 ;

EfY^2 gˆEf… 2 X‡ 1 †^2 gˆ 4 EfX^2 g‡ 4 EfXg‡ 1 ;

ˆ

EfYgˆEfX^2 gˆ

1

… 2 †^1 =^2

Z 1

1

x^2 e^ x

(^2) = 2
dxˆ 1 ; … 5 : 37 †
EfY^2 gˆEfX^4 gˆ

1

… 2 †^1 =^2

Z 1

1

x^4 e^ x

(^2) = 2
dxˆ 3 : … 5 : 38 †
^2 YˆEfY^2 g
E^2 fYgˆ 3
1 ˆ 2 : … 5 : 39 †