Fundamentals of Probability and Statistics for Engineers

H ence, Equation (5.67) leads to

It is of interest to note that the result given by Equation (5.72) can be written as

where

implying that, although Y 1 and Y 2 are both functions of X 1 and X 2 , they are
independent and identically and normally distributed.

Ex ample 5. 19. Problem: for the same distributions assigned to X 1 and X 2 in
Example 5.18, determine the jpdf of Y 1 (X 12 X 2 2 )1/2 and Y 2 X 1 /X 2.
Answer: let us first note that Y 1 takes values only in the positive range.
H ence,

For y 1 0, the transformationy g(x) admits two solutions. They are:

and

Functions of Random Variables 151

fY 1 Y 2 …y 1 ;y 2 †ˆfX 1 ‰g^11 …y†ŠfX 2 ‰g^21 …y†ŠjJj

ˆ

1

4 

exp



…y 1 ‡y 2 †^2

8

"

exp



…y 1

 y 2 †^2

8

"

ˆ

1

4 

exp

…y^21 ‡y^22 †

4



; …

1;

1†<…y 1 ;y 2 †<…1;1†:… 5 : 72 †

fY 1 Y 2 …y 1 ;y 2 †ˆfY 1 …y 1 †fY 2 …y 2 †; … 5 : 73 †

fY 1 …y 1 †ˆ

1

… 4 †^1 =^2

exp

y^21

4



; 

1 <y 1 < 1 ;

fY 2 …y 2 †ˆ

1

… 4 †^1 =^2

exp

y^22

4



; 

1 <y 2 < 1 ;

ˆ‡ ˆ

fY 1 Y 2 …y 1 ;y 2 †ˆ 0 ; y 1 < 0 :

 ˆ

x 11 ˆg^111 …y†ˆ

y 1 y 2

… 1 ‡y^22 †^1 =^2

;

x 12 ˆg^121 …y†ˆ

y 1

… 1 ‡y^22 †^1 =^2

;

x 21 ˆg^211 …y†ˆ

x 11 ;

x 22 ˆg^221 …y†ˆ

x 12 :