Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 161

From here, we have:

P ðX 1 X 2 Þ2A f g ¼ Pfx 1 a; x 2 cgPfx 1 a; x 2 bg: (3.24)

Using ( 3.9 ), ( 3.24 ) can be rewritten as:

P ðX 1 ; X 2 Þ2A f g ¼ F X 1 X 2 ða; cÞF X 1 X 2 ða; bÞ:

(3.25)

In general, for N random variables

ðX 1 ; X 2 ; ... ; X N Þ;

the joint distribution, denoted as F X 1 ; ... ; X N ðx 1 ; ... ; x N Þ , is then defined as:

F X 1 ; ... ; X N ðx 1 ; ... ; x N Þ¼PfX 1 x 1 ; ... ; X 1 x N g: (3.26)

3.2.2 Joint Density Function

The joint probability density function ( PDF )or joint density function or two-
dimensional density function or shortly, joint PDF , for pair of random variables
X 1 and X 2 , is denoted as, f X 1 X 2 ðx 1 ; x 2 Þ , and defined as:

f X 1 X 2 ðx 1 ; x 2 Þ¼ @

2 F X 1 X 2 ðx 1 ; x 2 Þ
@x 1 @x 2

: (3.27)

For discrete variables the derivations are not defined in the step discontinuities,
implying the introduction of delta functions at pairs of discrete points ( x 1 i , x 2 j ).
Therefore, the joint PDF is equal to (see [PEE93, pp. 358-359], [HEL91, p. 147]):
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 gdðx 1 x 1 i Þdðx 2 x 2 j Þ: (3.28)

j

Example 3.2.5 We can find the joint density functions for Examples 3.2.1, 3.2.2,
and 3.2.3. Using the distribution ( 3.18 ), and ( 3.27 ), we have:

f X 1 X 2 ðx 1 ; x 2 Þ¼þp 1 p 2 dðx 1 1 Þdðx 2 1 Þþp 1 ð 1 p 2 Þdðx 1 1 Þdðx 2 Þ
þð 1 p 1 Þp 2 dðx 1 Þdðx 2 1 Þþð 1 p 1 Þð 1 p 2 Þdðx 1 Þdðx 2 Þ:

(3.29)

Similarly, from ( 3.20 ) we have:

f X 1 X 2 ðx 1 ; x 2 Þ¼þ 1 ½

þdðx 1 Þdðx 2 1 Þþdðx 1 Þdðx 2 Þ:

= 4 dðx 1 1 Þdðx 2 1 Þþdðx 1 1 Þdðx 2 Þ

(3.30)