Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

144 Multivariate Distributions

together with the inverse functions

x 1 =w 1 (y 1 ,y 2 ,...,yn),x 2 =w 2 (y 1 ,y 2 ,...,yn),...,xn=wn(y 1 ,y 2 ,...,yn)

define a one-to-one transformation that mapsSontoT in they 1 ,y 2 ,...,ynspace

and, hence, maps the subsetAofSonto a subsetBofT. Let the first partial

derivatives of the inverse functions be continuous and let thenbyndeterminant

(called the Jacobian)

J=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∂x 1

∂y 1

∂x 1

∂y 2 ···

∂x 1

∂yn

∂x 2

∂y 1

∂x 2

∂y 2 ···

∂x 2

∂yn

..

.

..

.

..

.

∂xn

∂y 1

∂xn

∂y 2 ···

∂xn

∂yn

∣∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

not be identically zero inT.Then

∫

···

∫

A

f(x 1 ,x 2 ,...,xn)dx 1 dx 2 ···dxn

=

∫

···

∫

B

f[w 1 (y 1 ,...,yn),w 2 (y 1 ,...,yn),...,wn(y 1 ,...,yn)]|J|dy 1 dy 2 ···dyn.

Whenever the conditions of this theorem are satisfied, we can determine the joint pdf
ofnfunctions ofnrandom variables. Appropriate changes of notation in Section
2.2 (to indicaten-space as opposed to 2-space) are all that are needed to show
that the joint pdf of the random variablesY 1 =u 1 (X 1 ,X 2 ,...,Xn), ..., Yn=
un(X 1 ,X 2 ,...,Xn), where the joint pdf ofX 1 ,...,Xnisf(x 1 ,...,xn), is given by

g(y 1 ,y 2 ,...,yn)=f[w 1 (y 1 ,...,yn),...,wn(y 1 ,...,yn)]|J|,

where (y 1 ,y 2 ,...,yn)∈T, and is zero elsewhere.

Example 2.7.1.LetX 1 ,X 2 ,X 3 have the joint pdf

f(x 1 ,x 2 ,x 3 )=

{

48 x 1 x 2 x 3 0 <x 1 <x 2 <x 3 < 1

0elsewhere.

(2.7.1)

IfY 1 =X 1 /X 2 ,Y 2 =X 2 /X 3 ,andY 3 =X 3 , then the inverse transformation is given
by
x 1 =y 1 y 2 y 3 ,x 2 =y 2 y 3 ,andx 3 =y 3.

The Jacobian is given by

J=

∣ ∣ ∣ ∣ ∣ ∣

y 2 y 3 y 1 y 3 y 1 y 2

0 y 3 y 2

001

∣ ∣ ∣ ∣ ∣ ∣

=y 2 y^23.

Moreover, inequalities defining the support are equivalent to

0 <y 1 y 2 y 3 ,y 1 y 2 y 3 <y 2 y 3 ,y 2 y 3 <y 3 ,andy 3 < 1 ,