Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 167

The desired conditional density is obtained using ( 3.50 ):

f X 1 X 2 ð x 1 ; x 2 Þ
f X 2 ðx 2 Þ

f X 1 x 1 jx 2 ð Þ¼ : (3.61)

Finally, placing ( 3.58 ) and ( 3.60 ) into ( 3.61 ), we have:

2 e x 1 ðlþx 2 Þ

x 1 ðl þ x 2 Þ for x 1 > 0

f X 1 x 1 jx 2 ð Þ¼ : (3.62)

0 otherwise

3.2.4 Independent Random Variables

The two random variables, X 1 and X 2 , are independent if the events

fX 1 x 1 g and fX 2 x 1 g

are independent for any value of x 1 and x 2 .
Using ( 1.105 ), we can write:

PfX 1 x 1 ; X 2 x 1 g¼PfX 1 x 1 gPfX 2 x 1 g: (3.63)

From ( 3.9 ) and ( 2.10 ), the joint distribution is:

F X 1 X 2 ðx 1 ; x 2 Þ¼F X 1 ðx 1 ÞF X 1 ðx 1 Þ: (3.64)

Similarly, for the joint density we have:

f X 1 X 2 ðx 1 ; x 2 Þ¼f X 1 ðx 1 Þ f X 1 ðx 1 Þ: (3.65)

Therefore, if the random variables X 1 and X 2 are independent, then their joint
distributions and joint PDFs are equal to the products of the marginal distributions
and densities, respectively.

Example 3.2.8 Determine whether or not the random variables X 1 and X 2 are
independent, if the joint density is given as:

1 = 2 for 0 < x 1 < 2 0 < x 2 < 1

;

f X 1 X 2 ðx 1 ; x 2 Þ¼ : (3.66)

0 otherwise

Solution The joint density ( 3.66 ) can be rewritten as:

f X 1 X 2 ðx 1 ; x 2 Þ¼f X 1 ðx 1 Þf X 2 ðx 2 Þ;

(3.67)