Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 218

f X 2 X 3 ðx 2 ; x 3 Þ¼f X 2 ðx 2 Þf X 3 ðx 3 Þ: (3.340)

Are the random variables X 1 , X 2 , and X 3 independent?
Q.3.9. Is it possible for joint random variables to be of different types? For
example, one discrete random variable and one continuous random
variable?
Q.3.10. When is the correlation equal to the covariance?
Q.3.11. Is the coefficient of correlation zero, if random variables X and Y are
related as in Y ¼ X 2 ?
Q.3.12. Why is the covariance not zero if the variables are correlated?

3.9 Answers

A.3.1. Yes. As illustrated in the following example, two discrete random variables
X 1 and X 2 have the following possible values:

ðX 1 ¼ 1 ; X 2 ¼ 1 Þ; ðX 1 ¼ 1 ; X 2 ¼ 0 Þ; ðX 1 ¼ 0 ; X 2 ¼ 1 Þ; ðX 1 ¼ 0 ; X 2 ¼ 0 Þ: (3.341)

We consider two cases:

ð a Þ PfX 1 ¼ 1 ;

PfX 1 ¼ 1 ; X 2 ¼ 0 g¼PfX 1 ¼ 0 ; X 2 ¼ 1 g¼ 3 = 8 :

; X 2 ¼ 1 g¼PfX 1 ¼ 0 ; X 2 ¼ 0 g¼ 1 = 8

(3.342)

ð b Þ PfX 1 ¼ 1 ;X 2 ¼ 1 g¼PfX 1 ¼ 0 ;X 2 ¼ 0 g¼PfX 1 ¼ 1 ;X 2 ¼ 0 g

¼PfX 1 ¼ 0

(3.343)

;X 2 ¼ 1 g¼ 1 = 4 :

The joint density function in case (a) is found to be:

f X 1 X 2 ðx 1 ; x 2 Þ¼ 1 = 8 dðx 1 1 Þdðx 2 1 Þþdðx 1 Þdðx 2 Þ ½

(3.344)

þ 3 = 8 dðx 1 1 Þdðx 2 Þþdðx 1 Þdðx 2 1 Þ ½ :

Similarly, in case (b), we have:

f X 1 X 2 ðx 1 ; x 2 Þ¼ 1 4 ½dðx 1 1 Þdðx 2 1 Þþdðx 1 Þdðx 2 Þþdðx 1 1 Þdðx 2 Þ

þ dðx 1 Þdðx 2 1 Þ:

=

(3.345)

Note that the joint PDFs ( 3.344 ) and ( 3.345 ) are different.
Next we find the marginal densities.