ECE 5510: Random Processes
Lecture Notes
Fall 2008
Lecture 17
Today: Random Processes: (1) Expectation (2) Correlation &
Covariance (3) Wide Sense Stationarity
1
Expectation of Random Processes
1.1
Expected Value and Correlation
Def’n:
Expected Value of a Random Process
The expected value of continuous time random process
X
(
t
) is the
deterministic function
μ
X
(
t
) =
E
X
(
t
)
[
X
(
t
)]
for the discretetime random process
X
n
,
μ
X
[
n
] =
E
X
n
[
X
n
]
Example: What is the expected value of a Poisson process?
Let Poisson process
X
(
t
) have arrival rate
λ
. We know that
μ
X
(
t
)
=
E
X
(
t
)
[
X
(
t
)] =
∞
summationdisplay
x
=0
x
(
λt
)
x
x
!
e

λt
=
e

λt
∞
summationdisplay
x
=0
x
(
λt
)
x
x
!
=
e

λt
∞
summationdisplay
x
=1
(
λt
)
x
(
x

1)!
=
(
λt
)
e

λt
∞
summationdisplay
x
=1
(
λt
)
x

1
(
x

1)!
=
(
λt
)
e

λt
∞
summationdisplay
y
=0
(
λt
)
y
(
y
)!
=
(
λt
)
e

λt
e
λt
=
λt
(1)
This is how we intuitively started deriving the Poisson process  we
said that it is the process in which on average we have
λ
arrivals
per unit time. Thus we’d certainly expect to see
λt
arrivals after a
time duration
t
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
ECE 5510 Fall 2008
2
Example: What is the expected value of
X
n
, the number
of successes in a Bernoulli process after
n
trials?
We know that
X
n
is Binomial, with mean
np
.
This is the mean
function, if we consider it to be a function of
n
:
μ
X
[
n
] =
E
X
[
X
n
] =
np
.
1.2
Autocovariance and Autocorrelation
These next two definitions are the most critical concepts of the rest
of the semester. Generally, for a timevarying signal, we often want
to know two things:
•
How to predict its future value. (It is not deterministic.) We
will use the ‘autocovariance’ to determine this.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Chen,R
 Variance, Autocorrelation, Stationary process, Random Processes

Click to edit the document details