Random Processes - Random Signals and Processes Primer with MATLAB - page 373

Fig. 6.5 Description of process in two time instants

6.2.3 Description of Process in n Points

The previous discussion can be generalized by observing a process in n points.
Obviously, in this way we can obtain a more detailed description of a process.
However, the resulting description will be highly complex.
By observing a process in n points,

t 1 , , t n , we get n random variables:

...

X 1 ¼ Xðt 1 Þ; X 2 ¼ Xðt 2 Þ; ... ; X n ¼ Xðt n Þ: (6.7)

We define a joint distribution function of n th order, as:

F X 1 X 2 :::X n ðx 1 ; x 2 ; ... ; x n ; t 1 ; t 2 ; ... ; t n Þ
¼ PXðt 1 Þx 1 ; Xðt 2 Þx 2 ; ... ; Xðt n Þxf g
¼ PfX 1 x 1 ; X 2 x 2 ; ... ; X n x n ; t 1 ; t 2 ; ... ; t n g:

(6.8)

The corresponding joint density function of n th order is given as:

n

f X 1 X 2 ; ... ;X n ðx 1 ; x 2 ; ... ; x n ; t 1 ; t 2 ;: ... t n Þ¼ @

F X 1 X 2 ð x 1 ; x 2 ; ... ; x n ; t 1 ; t 2 ; ... ; t n Þ
@x 1 @x 2 :::@x n

Px 1 < X ð t 1 Þ x 1 þ d x 1 ; x 2 < X ð t 2 Þ x 2 þ d x 2 ; ... ; x n < X ð t n Þ x n þ d x n ; t 1 ; t 2 ; ... ; t n f g

:

¼

d x 1 d x 2 ; ... ; d x n

P f x 1 < X 1 x 1 þ d x 1 ; x 2 < X 2 x 2 þ d x 2 ;: ... x n < X n x n þ d x n ; t 1 ; t 2 ; ... ; t n g
d x 1 d x 2

¼

; ... ; d x n