Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.2. Transformations: Bivariate Random Variables 103

SinceBis arbitrary, the last integrand must be the joint pdf of (Y 1 ,Y 2 ). That is

the pdf of (Y 1 ,Y 2 )is

fY 1 ,Y 2 (y 1 ,y 2 )=

{

fX 1 ,X 2 [w 1 (y 1 ,y 2 ),w 2 (y 1 ,y 2 )]|J| (y 1 ,y 2 )∈T

0elsewhere.

(2.2.3)

Several examples of this result are given next.

x 2

x 1

y 2

y 1

S

T

A

B

Figure 2.2.1:A general sketch of the supports of (X 1 ,X 2 ), (S), and (Y 1 ,Y 2 ), (T).

Example 2.2.3.Reconsider Example 2.2.2, where (X 1 ,X 2 ) have the uniform dis-
tribution over the unit square with the pdf given in expression (2.2.1). The support
of (X 1 ,X 2 )isthesetS={(x 1 ,x 2 ):0<x 1 < 1 , 0 <x 2 < 1 }as depicted in Figure
2.2.2.

SupposeY 1 =X 1 +X 2 andY 2 =X 1 −X 2. The transformation is given by

y 1 =u 1 (x 1 ,x 2 )=x 1 +x 2

y 2 =u 2 (x 1 ,x 2 )=x 1 −x 2.

This transformation is one-to-one. We first determine the setT in they 1 y 2 -plane
that is the mapping ofSunder this transformation. Now

x 1 =w 1 (y 1 ,y 2 )=^12 (y 1 +y 2 )

x 2 =w 2 (y 1 ,y 2 )=^12 (y 1 −y 2 ).

To determine the setSin they 1 y 2 -plane onto whichT is mapped under the transfor-
mation, note that the boundaries ofSare transformed as follows into the boundaries