Page 154 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 137

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 f (y). One can determine f (y) before moment calculations but it
Y
Y
is less expedient when only moments of Y are desired. Another remark to be
made is that, since the transformation is linear (Y ˆ 2X ‡ 1) in this case, only
the first two moments of X are needed in finding the first two moments of Y ,
that is,

EfYgˆ Ef2X ‡ 1gˆ 2EfXg‡ 1;
2
2
2
EfY gˆ Ef 2X ‡ 1† gˆ 4EfX g‡ 4EfXg‡ 1;
as seen from Equations (5.34) and (5.35). When the transformation is nonlinear,
however, moments of X of different orders will be needed, as shown below.

Example 5.10. Problem: from Example 5.7, determine the mean and variance
2
of Y ˆ X . The mean of Y is, in terms of f (x),
X
1 Z 1 2
x =2
2
2
EfYgˆ EfX gˆ 1=2 x e dx ˆ 1; 5:37†
2 †
1
and the second moment of Y is given by
1 Z 1 4
x =2
2
2
4
EfY gˆ EfX gˆ 1=2 x e dx ˆ 3: 5:38†
2 †
1
Thus,
2
2
2
ˆ EfY g
E fYgˆ 3
1 ˆ 2: 5:39†
Y
In this case, complete knowledge of f (x) is not needed but we to need to
X
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 ,..., X n . 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 X n
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

TLFeBOOK