Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 160

Example 3.2.2 Find the joint distribution in Example 3.1.2.

Solution All outcomes have the same probability of occurrence, resulting in:

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

¼ PfX 1 ¼ 0 ; X 2 ¼ 1 g¼ 1 = 4 : (3.19)

From ( 3.19 ), we have:

F X 1 X 2 ðx 1 ; x 2 Þ¼þ 1 4 ½uðx 1 1 Þuðx 2 1 Þþuðx 1 1 Þuðx 2 Þþuðx 1 Þuðx 2 1 Þ

þ uðx 1 Þuðx 2 Þ:

=

(3.20)

Example 3.2.3 Find the joint distribution in Example 3.1.3.
Solution All outcomes have the same probability of occurrence. However, the
probabilities of the values of the random variables are:

Pfx 1 ¼ 1 ; x 2 ¼ 1 g¼Pf H ;

T gþPf T

;

H g¼ 1

=

4 þ 1

=

4 ¼ 1

= ;

2

Pfx 1 ¼ 1 ; x 2 ¼ 0 g¼Pf H ; H g¼ 1 = 4 ;
Pfx 1 ¼ 0 ; x 2 ¼ 1 g¼Pf T ; T g¼ 1 = 4 : (3.21)

The joint distribution is:

F X 1 X 2 ðx 1 ; x 2 Þ¼ 1 = 2 uðx 1 1 Þuðx 2 1 Þþ 1 = 4 uðx 1 1 Þuðx 2 Þþ 1 = 4 uðx 1 Þuðx 2 1 Þ:

(3.22)

Example 3.2.4 Find the probability that a random point ( x 1 , x 2 ) will fall in the area
A , as shown in Fig. 3.6 .

Solution Area A is defined as:

fx 1 a; x 2 cgfx 1 a; x 2 bg: (3.23)

Fig. 3.6 Area A in Example 3.2.4