Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.7. Transformations for Several Random Variables 145

which reduces to the supportT ofY 1 ,Y 2 ,Y 3 of

T ={(y 1 ,y 2 ,y 3 ): 0<yi< 1 ,i=1, 2 , 3 }.

Hence the joint pdf ofY 1 ,Y 2 ,Y 3 is

g(y 1 ,y 2 ,y 3 ) = 48(y 1 y 2 y 3 )(y 2 y 3 )y 3 |y 2 y 32 |

=

{

48 y 1 y 23 y^530 <yi< 1 ,i=1, 2 , 3

0elsewhere.

(2.7.2)

The marginal pdfs are

g 1 (y 1 )=2y 1 , 0 <y 1 < 1 ,zero elsewhere

g 2 (y 2 )=4y 23 , 0 <y 2 < 1 ,zero elsewhere

g 3 (y 3 )=6y 35 , 0 <y 3 < 1 ,zero elsewhere.

Becauseg(y 1 ,y 2 ,y 3 )=g 1 (y 1 )g 2 (y 2 )g 3 (y 3 ), the random variablesY 1 ,Y 2 ,Y 3 are mu-
tually independent.

Example 2.7.2.LetX 1 ,X 2 ,X 3 be iid with common pdf

f(x)=

{

e−x 0 <x<∞

0elsewhere.

Consequently, the joint pdf ofX 1 ,X 2 ,X 3 is

fX 1 ,X 2 ,X 3 (x 1 ,x 2 ,x 3 )=

{

e−

P 3

i=1xi 0 <xi<∞,i=1, 2 , 3

0elsewhere.

Consider the random variablesY 1 ,Y 2 ,Y 3 defined by

Y 1 =X 1 +XX^12 +X 3 ,Y 2 =X 1 +XX^22 +X 3 ,andY 3 =X 1 +X 2 +X 3.

Hence, the inverse transformation is given by

x 1 =y 1 y 3 ,x 2 =y 2 y 3 ,andx 3 =y 3 −y 1 y 3 −y 2 y 3 ,

with the Jacobian

J=

∣∣

∣

∣

∣

∣

y 3 0 y 1

0 y 3 y 2

−y 3 −y 3 1 −y 1 −y 2

∣∣

∣

∣

∣

∣

=y 32.

The support ofX 1 ,X 2 ,X 3 maps onto

0 <y 1 y 3 <∞, 0 <y 2 y 3 <∞,and 0<y 3 (1−y 1 −y 2 )<∞,

which is equivalent to the supportT given by

T ={(y 1 ,y 2 ,y 3 ): 0<y 1 , 0 <y 2 , 0 < 1 −y 1 −y 2 , 0 <y 3 <∞}.