Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 159

F X 1 X 2 ðx 1 ; 1Þ ¼ PfX 1 x 1 ; X 2 1g ¼ 0 : (3.11c)

P.3 F X 1 X 2 ð1;1Þ ¼ PfX 1 1 ; X 2 1g¼ 1 : (3.12)

P.4 Two-dimensional distribution is a nondecreasing function of both x 1 and x 2 .
P.5

F X 1 X 2 ðx 1 ;1Þ ¼ PfX 1 x 1 ; X 2 1g¼F X 1 ðx 1 Þ: (3.13)

F X 1 X 2 ð1; x 2 Þ¼PfX 1 1 ; X 2 x 2 g¼F X 2 ðx 2 Þ: (3.14)

The one-dimensional distributions in ( 3.13 ) and ( 3.14 ) are called marginal
distributions .
Next, we express the probability that the pair of variables X 1 and X 2 is in a given
space, in term of its two-dimensional distribution function :

Pfx 11 < X 1 x 12 ; x 21 < X 2 x 22 g¼Pfx 11 < X 1 x 12 ; X 2 x 22 g
Pfx 11 < X 1 x 12 ; X 2 < x 21 g¼PfX 1 x 12 ; X 2 x 22 gPfX 1 < x 11 ; X 2 x 22 g
PfX 1 x 12 ; X 2 < x 21 gPfX 1 < x 11 ; X 2 < x 21 ½
¼ F X 1 X 2 ðx 12 ; x 22 ÞF X 1 X 2 ðx 11 ; x 22 ÞF X 1 X 2 ðx 12 ; x 21 ÞþF X 1 X 2 ðx 11 ; x 21 Þ:

(3.15)

If a two-dimensional variable is discrete, then its distribution has two-dimensional
unit step functions u ( x 1 i , x 2 j ) ¼ u ( x 1 i ) u ( x 2 j ) in the corresponding discrete points
( x 1 i , x 2 j ), as shown in the following equation (see [PEE93, pp. 358-359]):
F X 1 X 2 ðx 1 ; x 2 Þ¼ X
i

X

PfX 1 ¼ x 1 i ;X 2 ¼ x 2 j guðx 1 x 1 i Þuðx 2 x 2 j Þ: (3.16)

j

Example 3.2.1 Find the joint distribution function from Example 3.1.1 considering
that the probability that the first and second messages are correct (denoted as p 1 and
p 2 , respectively), and that they are independent.

Pfx 1 ¼ 1 ; x 2 ¼ 1 g¼Pfx 1 ¼ 1 gPfx 2 ¼ 1 g¼p 1 p 2 ;

Pfx 1 ¼ 1 ; x 2 ¼ 0 g¼Pfx 1 ¼ 1 gPfx 2 ¼ 0 g¼p 1 ð 1 p 2 Þ;

Pfx 1 ¼ 0 ; x 2 ¼ 1 g¼Pfx 1 ¼ 0 gPfx 2 ¼ 1 g¼p 2 ð 1 p 1 Þ;

Pfx 1 ¼ 0 ; x 2 ¼ 0 g¼Pfx 1 ¼ 0 gPfx 2 ¼ 0 g¼ð 1 p 1 Þð 1 p 2 Þ: (3.17)

The joint distribution is:

F X 1 X 2 ðx 1 ; x 2 Þ¼þp 1 p 2 uðx 1 1 Þuðx 2 1 Þþp 1 ð 1 p 2 Þuðx 1 1 Þuðx 2 Þ
þð 1 p 1 Þp 2 uðx 1 Þuðx 2 1 Þþð 1 p 1 Þð 1 p 2 Þuðx 1 Þuðx 2 Þ: (3.18)