EE5620 Stochastic Processes
1
EE5620  Homework #4
Due on Friday, 11/24/2006, in class
Reading assignments:
Section 4.1
∼
Section 4.3, Gallager’s note.
Problem assignments:
1. (20 points) Suppose you want to estimate a random vector
x
from observations
y
. You know the information about the first and second order statistics of the
joint distribution of [
x
T
,
y
T
]
T
, but you don’t know the actual distribution itself.
You want to design a robust estimator, one which has decent performance no
matter what the joint distribution is. More precisely, for any estimator
φ
, let
its worstcase performance
e
(
φ
) be defined as:
e
(
φ
)
,
max
f
x
,
y
E
£

x

φ
(
y
)

2
/
,
where the maximum is taken over all joint distribution
f
with the given first and
second order statistics.
Find the estimator which has the optimal worstcase
performance. Justify your answer carefully.
2. (10+10+10=30 points) Consider a communication link with received signal
given by
Y
=
X
+
S
+
Z,
where
X
is the transmitted signal with mean
μ
and variance
σ
2
X
,
S
is a known
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 Winter '10
 GFung
 Normal Distribution, Probability theory, Estimation theory, joint distribution

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