Page 156 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 139

x
2

x x = y
1 2

x 1
R 2

2

Figure 5.16 Region R , in Example 5.11

Z 1 Z y=x 2 Z 0 Z 1
F Y y†ˆ f x 1 ; x 2 †dx 1 dx 2 ‡ f x 1 ; x 2 †dx 1 dx 2 : 5:44†
X 1 X 2 X 1 X 2
0
1
1 y=x 2
Substituting f (x 1 , x 2 ) into Equation (5.44) enables us to determine F Y (y)
X 1 X 2
and, on differentiating with respect to y, gives f (y).
Y
For the special case where X 1 and X 2 are independent, we have
f x 1 , x 2 ) ˆ f x 1 )f x 2 ); and Equation (5.44) simplifies to
X 1 X 2 X 1 X 2
1 0
Z y Z y
F Y y†ˆ F X 1 f x 2 †dx 2 ‡ 1
F X 1 f x 2 †dx 2 ;
X 2 X 2
0 x 2
1 x 2
and

y
dF Y y† Z 1

1
f y†ˆ ˆ f f x 2 † dx 2 : 5:45†
Y
dy
1 X 1 x 2 X 2
x 2

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