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ECE 154B Winter 2007
Homework #2 Solutions
1. Consider the 2 hypotheses problem where the a priori probabilities for the two
hypotheses are:
π
0
=
¾
and
π
1
=
¼
.
Let the conditional probability densities of
the observable Y under the two hypotheses be:
f
Y0
(y) = K
1
y for 0 <
y <
3,
f
Y0
(y) = 0 elsewhere
and
f
Y1
(y) = K
2
for 0 <
y <
3,
f
Y1
(y) = 0 elsewhere.
(You must first determine the constants K
1
and K
2.
).
a. Find the decision rule that minimizes the probability of error.
K
1
=2/9 (from Homework 1). K
2
=
1/3.
0
f
Y
∣
0
y
=
1
f
Y
∣
1
y
3
4
2
9
y
=
1
4
1
3
y
=
1
2
Choose H
0
if y > 1/2.
Choose H
1
if y < 1/2.
b. Calculate the probability of error for this decision rule.
P
error
=
0
P
error
∣
H
0
1
P
error
∣
H
1
=
3
4
∫
0
1
2
2
9
ydy
1
4
∫
1
2
3
1
3
dy
=
1
48
5
24
=
11
48
c. Verify your calculation of the probability of error by simulation.
Matlab:
x=rand(1,10000)>(3/4);
for n=1:10000
if x(n)==0
y(n)=3*sqrt(rand);
else
y(n)=3*rand;
end
end
z=y<(1/2);
sum(x~=z)/10000
2. Consider the 4 hypotheses problem with 4 equally likely hypotheses (H
0
, H
1
, H
2
,
H
3
) .
Under each of the 4 hypotheses assume that the observable Y is a Gaussian
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View Full Documentrandom variable with means (3, 1, +1, +3) respectively.
Under each hypotheses
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 Winter '07
 WOlf

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