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Kyung Hee University
Department of Electronics and Radio Engineering
C1002900
Random Processing
Homework 5
Spring 2010
Professor Hyundong Shin
Issued:
May 17, 2010
Due:
May 31, 2010
(No acceptance of overdue submission)
Reading:
Course textbook Chapter 8, Lecture Note X
HW 5.1
Suppose
X
and
Y
are random variables whose joint density, depicted in Fig. 5.1, is constant
in the shaded area and 0 elsewhere.
Figure 5.1
(a)
Let
H
H
0
when
X
0 , and
H
H
1
when
X
0 . Determine
P
HH
00
±
and
P
11
. Make fully labeled sketches of
YH
f
yH
0
and
f
1
.
(b)
Design a rule
ˆ
H y
for deciding between
H
0
and
H
1
given an observation
Y
y
, which
minimizes the probability of error. Specify for which values of
Y
your rule chooses
H
1
,
and for which values it chooses
H
0
. That is, determine the regions
ˆ
ˆ
.
yH y H
What is the resulting probability of error?
(c)
Sketch the operating characteristic of the likelihood ratio test (LLT) for this problem in the
DF
,
P
P
plane. On this plot, indicate also the region consisting of every
,
P
P
value that
can be achieved using some decision rule.
,
,
XY
fx
y
y
x
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(d)
Is the point corresponding to
D
P
23 and
F
P
56 in this region? If so, describe a test that
achieves this value. If not, explain.
HW 5.2
In the binary communication system shown in Fig. 5.2, messages
X
0 and
X
1 occur with
a
priori
probabilities 1/4 and 3/4, respectively. Suppose that we observe
YXW
where
,
W
3434
is statistically independent of whether message
X
0 or
X
1 occurs.
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 Spring '10
 HyungdongShin
 Probability, Probability theory, Kyung Hee University Department of Electronics, Hyundong Shin

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