ECE 5510: Random Processes
Lecture Notes
Fall 2008
Lecture 14
Today: (1) Decorrelation Transform (continued) (2) Joint Gaus
sian R.V.s
•
HW 6 is due today; HW 7 is due a week from today, at 5pm.
•
Discussion item?
0.1
Linear Transforms of R.V.s Continued
Example:
Average and Difference of Two i.i.d. R.V.s
We represent the arrival of two people for a meeting as r.v.s
X
1
and
X
2
.
Let
X
= [
X
1
, X
2
]
T
.
Assume that the two people arrive
independently, with the same variance
σ
2
and mean
μ
. Consider the
average arrival time
Y
1
= (
X
1
+
X
2
)
/
2, and the difference between
the arrival times,
Y
2
=
X
1

X
2
. The latter is a wait time that one
person must wait before the second person arrives. Show that the
average time and the difference between the times are uncorrelated.
You can take these steps to solve this problem:
1. Let
Y
= [
Y
1
, Y
2
]
T
.
What is the transform matrix
A
in the
relation
Y
=
A
X
?
2. What is the mean matrix
μ
Y
=
E
Y
[
Y
]? (Note this isn’t really
needed to answer the question, but is good practice anyway.)
3. What is the covariance matrix of
Y
?
4. How does the covariance matrix show that the two are uncor
related?
1
Joint Gaussian r.v.s
We often (OFTEN) see joint Gaussian r.v.s. E.g. ECE 5520, Digi
tal Communications, joint Gaussian r.v.s are everywhere. In addi
tion, the joint Gaussian R.V. is extremely important in statistics,
economics, other areas of engineering. We can’t overemphasize its
importance.
In many areas of the sciences, the Gaussian r.v.
is
an approximation. In digital communications, control systems, and
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 Fall '08
 Chen,R
 Covariance, Variance, Covariance matrix, multivariate gaussian

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