Digital Signal Processing Reference
In-Depth Information
time instant
t
1
is a random variable
X
1
. The distribution function of the variable,
denoted as
F
X
1
ðx
1
; t
1
Þ
is defined as:
F
X
1
ðx
1
; t
1
Þ¼PfX
1
x
1
;
t
1
g:
(6.1)
Note that the definition (
6.1
) is the same as the definition of distribution of a
random variable, with the only difference being that the distribution (
6.1
) depends
on the time instant
t
1
.
The meaning of the distribution (
6.1
) is explained in Fig.
6.4
. For the specified
time instant
t
1
and value
x
1
, the distribution (
6.1
) presents the probability that in the
instant
t
1
all realizations are less than or equal to a value
x
1
. This probability can
also be interpreted using a frequency ratio, as described below.
Consider that a random experiment is performed
n
times and a particular
realization is given to each outcome, as shown in Fig.
6.4
. Let
n
(
x
1
,
t
1
) be the total
number of successes where the amplitudes of realizations in the time instant
t
1
are
not more than
x
1
.
Taking
n
to be very large, the desired probability can be approximated as:
n
ð
x
1
;
t
1
Þ
n
F
X
1
ðx
1
; t
1
j
Þ
:
(6.2)
The distribution (
6.2
)iscalleda
distribution of the first order
,ora
one-dimensional
distribution
of a process
X
(
t
). Note that the one-dimensional distribution is obtained by
observing a process in one time instant.
Fig. 6.4
Description of a process in a particular time instant
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