Law Of Total Probability, Bayes' Formula And Binary Hypothesis Testing Worksheet With Answers - University Of Illinois, 2012 Page 5
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1
2
3
4
5
6
7π
0
Λ(k)
π
1
k 1
p
(1
p
)
π
1
1
0
k 1
p
(1
p
)
π
0
0
1
k 1
p
(1
p
)
π
1
1
0
ln
ln
k 1
p
(1
p
)
π
0
0
1
p
1
p
π
1
1
0
ln
+ (k
1) ln
ln
p
1
p
π
0
0
1
p
π
ln
0
ln
1
( )
π
p
1
0
k
+ 1
1 p
ln
1
1 p
0
π
p
0
0
ln
π
p
1
1
k
1 +
1 p
1
ln
1 p
0
1 p
where again ( ) follows because p
< p
so that ln
1
< 0.
0
1
1 p
0
1
1
(d) Find p
, p
, and p
under the MAP decision rule assuming p
= e
, p
= 1
e
.
M D
F A
e
0
1
Solution: Notice that with the given values of p
and p
, the MAP rule becomes ruling
0
1
ln 3 ln 2
in favor of H
if X
2 +
1.251. Therefore, MAP rules in favor of H
if
1
1
1
1 ln(1 e
)
X
1. Hence,
p
= P declare H
true H
= P X > 1 H
= 1
P X
1 H
M D
0
1
1
1
1 1
1
1
= 1
p
(1
p
)
= 1
(1
e
) = e
0.3679
1
1
1 1
1
p
= P declare H
true H
= P X
1 H
= p
(1
p
)
= e
0.3679
F A
1
0
0
0
0
And,
1
1
p
= π
p
+ π
p
= e
(π
+ π
) = e
0.3679
e
0
F A
1
M D
0
1
As expected, p
for the MAP rule is less than or equal to p
for the ML rule.
e
e
5. [Telephone game]
Suppose that in a game of telephone, the message to be passed on (sent) can be one of three
equally likely messages (m
, m
, m
). As the message passes from one end to the other, the
1
2
3
message can get distorted and the message received at the other end can be different than
the one sent. Suppose that if m
is sent, the conditional probabilities of the received message
1
being m
, m
, m
are 0.7, 0.2, 0.1, respectively. If m
is sent, the conditional probabilities
1
2
3
2
of the received message being m
, m
, m
are 0.05, 0.9, 0.05, respectively. If m
is sent, the
1
2
3
3
conditional probabilities of the received message being m
, m
, m
are 0.15, 0.05, 0.8, respec-
1
2
3
tively.
(a) For i = 1, 2, 3, find the probability that message m
is received.
i
Solution: For i = 0, 1, 2, let S
denote the event that message m
is sent, and let R
i
i
i
denote the event that message m
is received. Using the law of total probability we obtain
i
5
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