UCSD ECE153
Handout #26
Prof. YoungHan Kim
Thursday, May 19, 2011
Homework Set #6
Due: Thursday, May 26, 2011
1.
Covariance matrices.
Which of the following matrices can be a covariance matrix?
Justify your answer either by constructing a random vector
X
, as a function of the
i.i.d zero mean unit variance random variables
Z
1
, Z
2
,
and
Z
3
, with the given covariance
matrix, or by establishing a contradiction.
(a)
bracketleftbigg
1
2
0
2
bracketrightbigg
(b)
bracketleftbigg
2
1
1
2
bracketrightbigg
(c)
1
1
1
1
2
2
1
2
3
(d)
1
1
2
1
2
3
2
3
3
2.
Gaussian random vector.
Given a Gaussian random vector
X
∼ N
(
μ
,
Σ), where
μ
= (1 5 2)
T
and
Σ =
1
1
0
1
4
0
0
0
9
.
(a) Find the pdfs of
i.
X
1
,
ii.
X
2
+
X
3
,
iii. 2
X
1
+
X
2
+
X
3
,
iv.
X
3
given (
X
1
, X
2
), and
v. (
X
2
, X
3
) given
X
1
.
(b) What is
P
{
2
X
1
+
X
2

X
3
<
0
}
? Express your answer using the
Q
function.
(c) Find the joint pdf on
Y
=
A
X
, where
A
=
bracketleftbigg
2
1
1
1

1
1
bracketrightbigg
.
3.
Gaussian Markov chain.
Let
X, Y,
and
Z
be jointly Gaussian random variables with
zero mean and unit variance, i.e.,
E
(
X
) =
E
(
Y
) =
E
(
Z
) = 0 and
E
(
X
2
) =
E
(
Y
2
) =
E
(
Z
2
) = 1.
Let
ρ
X,Y
denote the correlation coefficient between
X
and
Y
, and let
ρ
Y,Z
denote the correlation coefficient between
Y
and
Z
. Suppose that
X
and
Z
are
conditionally independent given
Y
.
(a) Find
ρ
X,Z
in terms of
ρ
X,Y
and
ρ
Y,Z
.
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 Spring '11
 Kim
 Variance, Probability theory, Gaussian random vector

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