Fundamentals of Probability and Statistics for Engineers

where r is the number of solutions for x of equation yg( x), and Jj is
defined by

In the above, gj 1 ,gj 2 ,...,and gjn are components of gj.

As we mentioned earlier, the results presented above can also be applied to
the case in which the dimension ofYis smaller than that ofX. Consider the
transformation represented in Equation (5.60) in which m < n. In order to use
the formulae developed above, we first augment the m-d imensional random
vector Y by another (n m) – dimensional random vector Z. The vector Z can
be constructed as a simple function ofXin the form

wherehsatisfies conditions of continuity and continuity in partial derivatives.
On combin ing Equations (5.60) and (5.70), we have now an n-random-variable
to n-random-variable transformation, and the jpdf of Y and Z can be obtained
by means of Equation (5.67) or Equation (5.68). The jpdf ofYalone is then
found through integration with respect to the components ofZ.

Ex ample 5. 18. Problem: let random variables X 1 and X 2 be independent and
identically and normally distributed according to

and similarly for X 2. Determine the jpdf of Y 1 X 1 X 2 ,andY 2 X 1 X 2.
Answer: Equation (5.67) applies in this case. The solutions of x 1 and x 2 in
terms of y 1 and y 2 are

The Jacobian in this case takes the form

150 Fundamentals of Probability and Statistics for Engineers

ˆ

Jjˆ

qg^ j 11

qy 1

qg^ j 11

qy 2



qg^ j 11

qyn

... ...

qg^ jn^1

qy 1

qg^ jn^1

qy 2



qg^ jn^1

qyn

(^)
(^)

… 5 : 69 †

Zˆh…X†; … 5 : 70 †

fX 1 …x 1 †ˆ

1

… 2 †^1 =^2

exp

x^21

2



; 

1 <x 1 < 1 ;

ˆ‡ ˆ

x 1 ˆg^11 …y†ˆ

y 1 ‡y 2

2

; x 2 ˆg^21 …y†ˆ

y 1

 y 2

2

: … 5 : 71 †

Jˆ

qg^11

qy 1

qg^11

qy 2

qg^21

qy 1

qg^21

qy 2

ˆ

1

2

1

2

1

2

1

2

ˆ

1

2

: