1
Kyung Hee University
Department of Electronics and Radio Engineering
C1002900
Random Processing
Homework 2
Spring 2010
Professor Hyundong Shin
Issued:
April 14, 2010
Due:
April 28, 2010
Reading:
Course textbook Chapter 7
HW 2.1
Consider the following
3
3
matrices:
.
A
B
C
D
E
F
G
10
3
1
10
5
2
10
5
2
10
5
2
2
5
0
5
3
3
5
3
3
5
3
1
1
0
2
2
3
2
2
3
2
2
1
2
10
5
2
10
0
0
2
1
1
5
3
1
0
3
0
1
2
1
2
1
2
0
0
2
1
1
2
Your answers to the following questions may consist of more than one of the above matrices or
none of them. Justify your answers.
(a)
Which of the above could be the covariance matrix of some random vector?
(b)
Which of the above could be the crosscovariance matrix of two random vectors?
(c)
Which of the above could be the covariance matrix of a random vector in which one com
ponent is a linear combination of the other two components?
(d)
Which of the above could be the covariance matrix of a random vector with statistically
independent components? Must a random vector with such a covariance matrix have sta
tistically independent components?
HW 2.2
For each of the following statements, determine whether it is TRUE or FALSE. Make sure you
give a sufficient but brief justification of your answers. Note that each statement is independent
of the others, so you cannot use the assumptions of one in the others.
(a)
If random vectors
X
and
Y
are jointly Gaussian and

E
E
X Y
X
, then
X
and
Y
are statistically independent.
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(b)
Given two random variables
X
and
Y
, if

E
X Y
E
X
, then for all functions
g
,
E
Xg Y
E
X
E g Y
.
(c)
Given two random variables
X
and
Y
, if
E
Xg Y
E
X
E g Y
for all functions
g
,
then

E
X Y
E
X
.
HW 2.3
We wish to estimate a random variable
X
by some constant
ˆ
x
. There are many ways to meas
ure how good an estimate
ˆ
x
is. Here you will derive an important property of the
minimum
mean square error estimation (MMSE)
. Define the
mean square estimation error
by
ˆ
ˆ
(
)
e x
E
X
x
2
.
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 Spring '10
 HyungdongShin
 Probability theory

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