Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 205

The joint density ( 3.294 ) can be rewritten as:

f Y 1 Y 2 ðy 1 ; y 2 Þ¼f Y 1 ðy 1 Þf Y 2 ðy 2 Þ; (3.295)

where

f Y 1 ðy 1 Þ¼y 1 e y 1 ; y 1 0 ;

for

f Y 2 ðy 2 Þ¼ 1 ; for 0 y 2 1 : (3.296)

Therefore, from ( 3.295 ) it follows that the random variables Y 1 and Y 2 are also
independent.

Exercise 3.13 The random variables X 1 and X 2 are independent and have the
following density functions:

e x 1 e x 2 for x 1 > 0 for x 2 > 0

f X 1 ðx 1 Þ¼

otherwise ;

f X 2 ðx 2 Þ¼ : (3.297)

0 0 otherwise

Find the PDF of the random variable X ,if

X 1
X 2 :

X ¼ (3.298)

Solution We have two input random variables and one output variable. In order to
apply the expression ( 3.159 ) we must first introduce the auxiliary variable Y ,

Y ¼ X 1 : (3.299)

Now we have the transformation defined in the following two equations:

x 1
x 2 ;
y ¼ x 1 :

x ¼

(3.300)

This system of equations ( 3.300 ) has one solution:

x 11 ¼ y; and x 21 ¼ y=x: (3.301)

The Jacobian of the transformation ( 3.300 ) is:

¼

1
x 2

x 1
x 2
10

x 2
y :

x 1
x 2 ¼

y

J ¼

=x 2 ¼

(3.302)

y 2