Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 207

Exercise 3.15 The random variables X 1 and X 2 are independent. Find the PDF of
the random variable X ,if

X 1
X 2 :

X ¼

(3.312)

The marginal densities are:

x 2
2 b 2

x 1
2 a 2 uðx 1 Þ;

b 2 e

x 1 x 2

a 2 e

f X 1 ðx 1 Þ¼ f X 2 ðx 2 Þ¼ uðx 2 Þ: (3.313)

Answer The auxiliary variable Y ¼ X 2 is introduced. The corresponding set of the
transformation equations

x ¼ x 1 =x 2 ;
y ¼ x 2 ; (3.314)

has the unique solution x 1 ¼ xy and x 2 ¼ y . The Jacobian of the transformation
( 3.314 )is

¼

x 1
x 2
01

x 2

1

1
x 2 ¼

1
y :

J ¼ (3.315)

The joint density function of the variables X and Y is given as:

x 1 x 2

b 2

y 2
2

x 2
a 2 þ

1
2

1
b 2

a 2 þ

a 2 b 2 e a 2 b 2 e

xy 3 f X 1 X 2 ð x ; y Þ

Jðx; yÞ

yx 1 x 2

f XY ðx; yÞ¼

j ¼

¼ : (3.316)

j

Denoting

x 2
a 2 þ

1
2

1
b 2

a ¼ ; (3.317)

from ( 3.316 ) and ( 3.317 ), we have:

1

a 2 b 2 1
0

x

y 3 e ay 2 d y:

f X ðxÞ¼ f XY ðx; yÞ d y ¼ (3.318)

0

Using the integral 4 from Appendix A and (E.2.15.6) we have:

2 a 2
b 2

x
a 2 b 2

1
2 a 2 ¼

x

f X ðxÞ¼ x 0 (3.319)

2 ;

a 2
b 2

x 2 þ