Page 159 - Fundamentals of Probability and Statistics for Engineers

142 Fundamentals of Probability and Statistics for Engineers

For independent X 1 and X 2 ,
Z 1 Z x 2 y
F Y y†ˆ f x 1 †f x 2 †dx 1 dx 2 ;
X 1 X 2
0 0
Z 1
ˆ F X 1 x 2 y†f x 2 †dx 2 ; for y > 0; 5:49†
X 2
0
ˆ 0; elsewhere:

The pdf of Y is thus given by, on differentiating Equation (5.49) with respect
to y,

8
Z 1
>
< x 2 f x 2 y†f x 2 †dx 2 ; for y > 0;
f y†ˆ 0 X 1 X 2 5:50†
Y
>
0; elsewhere;
:
and, on substituting Equation (5.48) into Equation (5.50), it takes the form
8 Z 1 1
x 2 e e dx 2 ˆ ; for y > 0;
>
x 2 y
x 2
< 2
f y†ˆ 0 1 ‡ y† 5:51†
Y
>
:
0; elsewhere:
Again, it is easy to check that
1
Z
f y†dy ˆ 1:
Y
0
Example 5.13. To show that it is elementary to obtain solutions to problems
discussed in this section when X 1 , X 2 ,..., and X n are discrete, consider again
Y ˆ X 1 /X 2 given that X 1 and X 2 are discrete and their joint probability mass
function (jpmf) p (x 1 , x 2 ) is tabulated in Table 5.1. In this case, the pmf of Y
X 1 X 2
is easily determined by assignment of probabilities p (x 1 , x 2 ) to the corres-
X 1 X 2
ponding values of y ˆ x 1 /x 2 . Thus, we obtain:
1
8
0:5; for y ˆ ;
>
>
>
> 2
>
0:24 ‡ 0:04 ˆ 0:28; for y ˆ 1;
>
>
>
<
p y†ˆ 3
Y 0:04; for y ˆ ;
>
> 2
>
>
0:06; for y ˆ 2;
>
>
>
>
0:12; for y ˆ 3:
:
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