Databases Reference
In-Depth Information
If all the random variables
X
1
,
are independent and they have the same distribution,
they are said to be
independent, identically distributed
(
iid
).
Two random variables
X
1
and
X
2
aresaidtobe
orthogonal
if
X
2
,...
E
[
X
1
X
2
]=
0
(A.26)
Two random variables
X
1
and
X
2
are said to be
uncorrelated
if
E
[
(
X
1
−
μ
1
)(
X
2
−
μ
2
)
]=
0
(A.27)
where
.
The
autocorrelation function
of a random process is defined as
μ
1
=
E
[
X
1
]
and
μ
2
=
E
[
X
2
]
R
xx
(
t
i
,
t
2
)
=
E
[
X
1
X
2
]
(A.28)
For a given value of
N
, suppose we sample the stochastic process at
N
times
{
t
i
}
to get the
N
random variables
{
X
i
}
with
cdf F
X
1
X
2
...
X
N
(
x
1
,
x
2
,...,
x
N
)
, and another
N
times
{
t
i
+
T
}
X
i
}
x
1
,
x
2
,...,
x
N
)
to get the random variables
{
with
cdf F
X
1
X
2
...
X
N
(
.If
x
1
,
x
2
,...,
x
N
)
F
X
1
X
2
...
X
N
(
x
1
,
x
2
,...,
x
N
)
=
F
X
1
X
2
...
X
N
(
(A.29)
for all
N
and
T
, the process is said to be
stationary
.
The assumption of stationarity is a rather important assumption because it is a statement that
the statistical characteristics of the process under investigation do not change with time. Thus,
if we design a system for an input based on the statistical characteristics of the input today, the
system will still be useful tomorrow because the input will not change its characteristics. The
assumption of stationarity is also a very strong assumption, and we can usually make do quite
well with a weaker condition,
wide sense
or
weak sense
stationarity.
A stochastic process is said to be wide sense or weak sense stationary if it satisfies the
following conditions:
1.
The mean is constant; that is,
μ(
t
)
=
μ
for all
t
.
2.
The variance is finite.
3.
The autocorrelation function
R
xx
(
is a function only of the difference between
t
1
and
t
2
, and not of the individual values of
t
1
and
t
2
; that is,
t
1
,
t
2
)
R
xx
(
t
1
,
t
2
)
=
R
xx
(
t
1
−
t
2
)
=
R
xx
(
t
2
−
t
1
).
(A.30)
Further Reading
1.
The classic topics on probability are the two-volume set
An Introduction to Probability
Theory and Its Applications,
by W. Feller [
287
,
288
].
2.
A commonly used text for an introductory course on probability and random processes
is
Probability, Random Variables, and Stochastic Processes
, by A. Papoulis [
289
].
