Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 158

Fig. 3.5 Mapping from the space S to the space ( x 1 , x 2 ) in Example 3.1.3

3.2 Joint Distribution and Density

3.2.1 Joint Distribution

The cumulative distribution function, joint distribution function or, shortly, joint
distribution of a pair of random variables X 1 and X 2 is defined as the probability of
joint events:

fX 1 x 1 ; X 2 x 2 g; (3.8)

and is denoted as F X 1 X 2 ðx 1 ; x 2 Þ .
Therefore, we have:

F X 1 X 2 ðx 1 ; x 2 Þ¼PfX 1 x 1 ; X 2 x 2 g; 1 < x 1 < 1;
1< x 2 < 1; (3.9)

where x 1 and x 2 are values within the two-dimensional space as shown in ( 3.9 ). The
joint distribution is also called two-dimensional distribution ,or second distribution .
In this context, distribution of one random variable is also called one-dimensional
distribution ,or first distribution .
From the properties of a one-dimensional distribution, we easily find the follow-
ing properties for a two-dimensional distribution:

0 F X 1 X 2 ðx 1 ; x 2 Þ 1 : (3.10)

P.1

F X 1 X 2 ð1; 1Þ ¼ PfX 1 1 ; X 2 1g ¼ 0 :

P.2 (3.11a)

F X 1 X 2 ð1; x 2 Þ¼PfX 1 1 ; X 2 x 2 g¼ 0 : (3.11b)