Page 168 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 151

Hence, Equation (5.67) leads to

1

1
f y 1 ; y 2 †ˆ f ‰g y†Šf ‰g y†ŠjJj
Y 1 Y 2 X 1 1 X 2 2
" 2 # " 2 #
1
y 1 ‡ y 2 †
y 1
y 2 †
ˆ exp exp
4 8 8
2
2
1
y ‡ y †
ˆ exp 1 2 ;
1;
1† < y 1 ; y 2 † < 1; 1†: 5:72†
4 4
It is of interest to note that the result given by Equation (5.72) can be written as
f y 1 ; y 2 †ˆ f y 1 †f y 2 †; 5:73†
Y 1 Y 2 Y 1 Y 2
where

1
y 2 1
f y 1 †ˆ exp ;
1 < y 1 < 1;
Y 1 1=2 4
4 †
1
y 2 2
f y 2 †ˆ exp ;
1 < y 2 < 1;
Y 2 1=2 4
4 †
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.

Example 5.19. Problem: for the same distributions assigned to X 1 and X 2 in
2 1/2
2
Example 5.18, determine the jpdf of Y 1 ˆ (X ‡ X ) and Y 2 ˆ X 1 /X 2 .
1 2
Answer: let us first note that Y 1 takes values only in the positive range.
Hence,
f y 1 ; y 2 †ˆ 0; y 1 < 0:
Y 1 Y 2
For y 1 0, the transformation y ˆ g(x) admits two solutions. They are:

1
x 11 ˆ g y†ˆ y 1 y 2 ;
11 2 1=2
1 ‡ y †
2

1
x 12 ˆ g y†ˆ y 1 ;
12 2 1=2
1 ‡ y †
2
and

1
x 21 ˆ g y†ˆ
x 11 ;
21

1
x 22 ˆ g y†ˆ
x 12 :
22

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