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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
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This homework help was uploaded on 02/11/2008 for the course ELEC 531 taught by Professor Lexa during the Fall '07 term at Rice.
 Fall '07
 Lexa
 Signal Processing

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