Fundamentals of Probability and Statistics for Engineers

In order to conform with conditions stated in this section, we augment Equa-
tion (5.78) by some simple transformation such as

The random variables Y and Z now play the role of Y 1 and Y 2 in Equation
(5.67) and we have

where

Using specific forms of fX 1 (x 1 )andfX 2 (x 2 ) given in Example (5.11), Equation
(5.80) becomes

Finally, pdf fY (y) is found by performing integration of Equation (5.81) with
respect to z:

This result agrees with that given in Equation (5.47) in Example 5.11.

Reference

Courant, R., 1937, Differential and Integral Calculus, Volume II, Wiley-Interscience,
New York.

Functions of Random Variables 153

ZˆX 2 : … 5 : 79 †

fYZ…y;z†ˆfX 1 X 2 ‰g^11 …y;z†;g^21 …y;z†ŠjJj;… 5 : 80 †

g^11 …y;z†ˆ

y

z;

g^21 …y;z†ˆz;

Jˆ

1

z^

y

z^2

01

(^)
(^)
ˆ^1 z:
fYZ…y;z†ˆfX 1
y
z



fX 2 …z†

1

z

ˆ

2 y

z



2

 z

2 z



;

ˆ

y… 2

 z†

z^2

; for 0y 2 ;andyz 2 ;

ˆ 0 ; elsewhere:

… 5 : 81 †

fY…y†ˆ

Z 1

1

fYZ…y;z†dzˆ

Z 2

y

y… 2

 z†

z^2



dz;

ˆ 2 ‡y…lny 

1

 ln 2†; for 0y 2 ;

ˆ 0 ; elsewhere: