Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

104 Multivariate Distributions

x 2

x 1 = 0x 1 = 1

x 2 = 1

(0, 0) x 2 = 0

x 1

S

Figure 2.2.2:The support of (X 1 ,X 2 ) of Example 2.2.3.

ofT:

x 1 =0 into 0=^12 (y 1 +y 2 )

x 1 =1 into 1=^12 (y 1 +y 2 )

x 2 =0 into 0=^12 (y 1 −y 2 )

x 2 =1 into 1=^12 (y 1 −y 2 ).

Accordingly,T is shown in Figure 2.2.3. Next, the Jacobian is given by

J=

∣ ∣ ∣ ∣ ∣ ∣ ∣

∂x 1

∂y 1

∂x 1

∂y 2

∂x 2

∂y 1

∂x 2

∂y 2

∣ ∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣

1

2

1

2

1

2 −

1

2

∣ ∣ ∣ ∣ ∣ ∣

=−

1

2

.

Although we suggest transforming the boundaries ofS, others might want to
use the inequalities
0 <x 1 <1and0<x 2 < 1
directly. These four inequalities become

0 <^12 (y 1 +y 2 )<1and0<^12 (y 1 −y 2 )< 1.

It is easy to see that these are equivalent to

−y 1 <y 2 ,y 2 < 2 −y 1 ,y 2 <y 1 y 1 − 2 <y 2 ;

and they define the setT.

Hence, the joint pdf of (Y 1 ,Y 2 )isgivenby

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

{

fX 1 ,X 2 [^12 (y 1 +y 2 ),^12 (y 1 −y 2 )]|J|=^12 (y 1 ,y 2 )∈T

0elsewhere.