EE 562a
Homework 3
Due Monday June 19, 2006
Work all 12 problems
.
Problem 1.
Suppose
X
is a meanzero random vector with covariance
matrix
K
X
=
⎡
⎢
⎣
1
4
3
4
20
22
3
22
36
⎤
⎥
⎦
.
Find a transformation
A
such that
Y
=
AX
has covariance matrix
K
Y
=
I
.
Problem 2.
Let
Z
(
n
) be an i.i.d. Bernoulli sequence where
P
(
Z
(
n
) = 1) =
p,
P
(
Z
(
n
) =
−
1) =
q
= 1
−
p.
Let
X
(
n
) =
n
k
=0
Z
(
k
)
where we take
Z
(0) = 0 =
X
(0). Then
X
(
n
) is a discrete random walk.
Find
R
X
(
n, m
) for this random walk.
1
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Problem 3.
Let
X
n
, n
≥
1, denote a sequence of independent and identi
cally distributed zeromean unitvariance Gaussian random variables.
Define for
n
≥
2
Y
n
=
1
2
(
X
n
+
X
n

1
)
.
a. Does this sequence converge in the mean square sense and, if so, to
what limit?
b. Does this sequence converge in distribution and, if so, to what distri
bution?
Problem 4.
Suppose the discrete random variable
U
takes on values
u
= 0
, u
=
1
2
, u
= 1
each with probability 1/3. Determine whether each of the following sequences
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 Summer '07
 ToddBrun
 Probability theory

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