Fundamentals of Probability and Statistics for Engineers

Substituting fX 1 X 2 (x 1 ,x 2 ) into Equation (5.44) enables us to determine FY (y)
and, on differentiating with respect to y, gives fY (y).
For the special case where X 1 and X 2 are independent, we have
and Equation (5.44) simplifies to

x 1 x 2 =y

x 1

x 2

R^2

Figure 5. 16 Region R^2 , in Example 5.11

Functions of Random Variables 139

FY…y†ˆ

Z 1

0

Zy=x 2

1

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2 ‡

Z 0

1

Z 1

y=x 2

fX 1 X 2 …x 1 ;x 2 †dx 1 dx 2 : … 5 : 44 †

fX 1 X 2 x 1 ,x 2 )ˆfX 1 x 1 )fX 2 x 2

FY…y†ˆ

Z 1

0

FX 1

y

x 2



fX 2 …x 2 †dx 2 ‡

Z 0

1

1

FX 1

y

x 2



fX 2 …x 2 †dx 2 ;

fY…y†ˆ

dFY…y†

dy

ˆ

Z 1

1

fX 1

y

x 2



fX 2 …x 2 †

1

x 2

dx^2 : …^5 :^45 †

and

);