Random Processes - Random Signals and Processes Primer with MATLAB - page 372

A density function is obtained as the derivation of the corresponding distribution
function,

f X 1 ðx 1 ; t 1 Þ¼ @ F X 1 ð x 1 ; t 1 Þ
@x 1

P f x 1 < X 1 x 1 þ d x 1 ; t 1 g
d x 1

¼ : (6.3)

The density function ( 6.3 ) is called a density function of the first order ,ora
one-dimensional density.

6.2.2 Description of a Process in Two Points

Next we observe a process in two points denoted as t 1 and t 2 . The corresponding
random variables are X 1 and X 2 .A joint distribution function ,a second order
distribution ,ora two-dimensional distribution is defined as:

F X 1 X 2 ðx 1 ; x 2 ; t 1 ; t 2 Þ¼PXðt 1 Þx 1 ; Xðt 2 Þx 2 f g

¼ PfX 1 x 1 ; X 2 x 2 ; t 1 ; t 2 g:

(6.4)

Note again that this is the same definition as that of a joint distribution of two
random variables with only exception being that now the joint distribution depends
on time instants t 1 and t 2 .
The meaning of this distribution can also be interpreted using a frequency ratio, as
shown in Fig. 6.5 . Consider a total number of outcomes for which the corresponding
realizations are not more than x 1 in a given time instant t 1 and also not more than x 2 in
a given time instant t 2 . The obtained number is denoted as n ( x 1 , x 2 , t 1 , t 2 ). Then,
assuming a large n , a distribution function can be expressed as:

n ð x 1 ; x 2 ; t 1 ; t 2 Þ
n

F X 1 X 2 ðx 1 ; x 2 ; t 1 ; t 2 Þ¼PfX 1 x 1 ; X 2 x 2 ; t 1 ; t 2 g : (6.5)

The corresponding density function is given as:

2 F X 1 X 2 ð x 1 ; x 2 ; t 1 ; t 2 Þ
@x 1 @x 2

f X 1 X 2 ðx 1 ; x 2 ; t 1 ; t 2 Þ¼ @

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 ; t 1 ; t 2 f g

¼

d x 1 d x 2

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

¼ : (6.6)