ECE 5510: Random Processes
Lecture Notes
Fall 2009
Lecture 11
Today: Joint r.v.s (1) Expectation of Joint r.v.s, (2) Covari
ance (both in Y&G 4.7)
•
HW 5 due Tue after break at 5pm in the HW locker. Office
hours: Today after class to 1:15, and Thu 13.
•
By the end of these lecture notes, we will have covered 4.14.5,
4.74.10; and 5.15.4, 5.6.
•
For the lecture 12, read 4.6 and 4.11, and 5.5.
1
Expectation of Joint r.v.s
We can find the expected value of a random variable or a function
of a random variable similar to how we learned it earlier. However,
now we have a more complex model, and the calculation may be
more complicated.
Def’n:
Expected Value (Joint)
The expected value of a function
g
(
X
1
, X
2
) is given by,
1. Discrete:
E
X
1
,X
2
[
g
(
X
1
, X
2
)]
=
∑
X
1
∈
S
X
1
∑
X
2
∈
S
X
2
g
(
X
1
, X
2
)
P
X
1
,X
2
(
x
1
, x
2
)
2. Continuous:
E
X
1
,X
2
[
g
(
X
1
, X
2
)]
=
integraltext
X
1
∈
S
X
1
integraltext
X
2
∈
S
X
2
g
(
X
1
, X
2
)
P
X
1
,X
2
(
x
1
, x
2
)
Essentially we have a function of two random variables
X
1
and
X
2
,
called
Y
=
g
(
X
1
, X
2
).
Remember when we had a function of a
random variable? We could either
1. Derive the pdf/pmf of Y, and then take the expected value of
Y
.
2. Take the expected value of
g
(
X
1
, X
2
) using directly the model
of
X
1
, X
2
.
Example: Expected Value of
g
(
X
1
)
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ECE 5510 Fall 2009
2
Let’s look at the discrete case:
E
X
1
,X
2
[
g
(
X
1
)]
=
summationdisplay
X
1
∈
S
X
1
summationdisplay
X
2
∈
S
X
2
g
(
X
1
)
P
X
1
,X
2
(
x
1
, x
2
)
=
summationdisplay
X
1
∈
S
X
1
g
(
X
1
)
summationdisplay
X
2
∈
S
X
2
P
X
1
,X
2
(
x
1
, x
2
)
=
summationdisplay
X
1
∈
S
X
1
g
(
X
1
)
P
X
1
(
x
1
)
=
E
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 Fall '08
 Chen,R
 Variance, Probability theory, X1

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