ELEC 531: STATISTICAL SIGNAL PROCESSING
Department of Electrical and Computer Engineering
Rice University
Fall 2007
Problem Set 7
Due: November 6, 2007
Problem 7.1
Consider the joint density of the random variables
X
and
Y
shown in Fig
ure 1 and the hypothesis test
H
0
:
X
≤
0
H
1
:
X >
0
.
The idea here is that we want to make a decision about the sign of the realization
X
=
x
based off the corresponding observation
Y
=
y
.
2
2
2
2
1
1
1
1
p
X,Y
(
x
,
y
)
x
y
Figure 1:
(a) Determine the prior probabilities
π
0
and
π
1
and make fully labeled sketches of the
conditional densities
p
Y

H
0
(
y

H
0
) and
p
Y

H
1
(
y

H
1
).
Assume the shaded region
in the Figure is constant while all other values of the joint density are zero.
(b) What is the decision rule that minimizes the probability of error? What is the
resulting probability of error?
(c) Sketch the receiver operating characteristic for this likelihood ratio test.
Problem 7.2
In the binary communications system shown below, messages
m
= 0 and
1
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m
= 1 occur with a priori probabilities
1
4
and
3
4
. When transmitted, these messages are
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 Fall '07
 Lexa
 Probability, Signal Processing, Probability theory, 1 L, 1 m, falsealarm probability

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