Page 162 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 145

We have not given examples in which functions of more than two random
variables are involved. Although more complicated problems can be formu-
lated in a similar fashion, it is in general more difficult to identify appropriate
n
regions R required by Equation (5.42), and the integrals are, of course, more
difficult to carry out. In principle, however, no intrinsic difficulties present
themselves in cases of functions of more than two random variables.

5.2.1 SUMS OF RANDOM VARIABLES

One of the most important transformations we encounter is a sum of random
variables. It has been discussed in Chapter 4 in the context of characteristic
functions. In fact, the technique of characteristic functions remains to be the
most powerful technique for sums of independent random variables.
In this section, the procedure presented in the above is used to give an
alternate method of attack.
Consider the sum
Y ˆ g X 1 ; ... ; X n †ˆ X 1 ‡ X 2 ‡ ‡ X n : 5:52†

It suffices to determine f (y) for n ˆ 2. The result for this case can then be
Y
applied successively to give the probability distribution of a sum of any number
of random variables. For Y ˆ X 1 ‡ X 2 , Equations (5.41) and (5.42) give
ZZ
F Y y†ˆ f x 1 ; x 2 †dx 1 dx 2 ;
X 1 X 2
2
R : x 1 ‡x 2 y†
and, as seen from Figure 5.20,
Z 1 Z y
x 2
F Y y†ˆ f x 1 ; x 2 †dx 1 dx 2 : 5:53†
X 1 X 2

1
1
Upon differentiating with respect to y we obtain

1
Z
f y†ˆ f y
x 2 ; x 2 †dx 2 : 5:54†
Y X 1 X 2

1
When X 1 and X 2 are independent, the above result further reduces to
Z 1
f y†ˆ f X 1 y
x 2 †f X 2 x 2 †dx 2 : 5:55†
Y

1
Integrals of the form given above arise often in practice. It is called convolution
of the functions f (x 1 ) and f (x 2 ).
X 1 X 2

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