Law Of Total Probability, Bayes' Formula And Binary Hypothesis Testing Worksheet With Answers - University Of Illinois, 2012 Page 3
ADVERTISEMENT
1
2
3
4
5
6
70
2 0
2
1
1
p
(0) = [1]
1
Y
0
2
2
0
1 0
1
2 1
1
1
1
2
1
1
+
1
1
0
13
13
1
2
2
0
2 0
2
2 2
2
1
1
2
1
1
625
+
1
1
=
0.9246
0
13
13
2
2
2
676
0
2 0
2
1
1
p
(1) = [0]
1
Y
0
2
2
1
1 1
1
2 1
1
1
1
2
1
1
+
1
1
1
13
13
1
2
2
1
2 1
2
2 2
2
1
1
2
1
1
50
+
1
1
=
0.074
1
13
13
2
2
2
676
0
2 0
1
2 1
2
1
1
2
1
1
p
(2) = [0]
1
+ [0]
1
Y
0
2
2
1
2
2
2
2 2
2
2 2
2
1
1
2
1
1
1
+
1
1
=
0.0015
2
13
13
2
2
2
676
(c) Find P X = 2 Y = 1 .
Solution: Using Bayes’ formula,
P Y = 1 X = 2 P X = 2
P X = 2 Y = 1
=
P Y = 1
1
2 1
2
2 2
2
2
1
1
1
1
1
1
16224
12
1
13
13
2
2
2
=
=
=
0.48
50
33800
25
576
4. [Calvin tries to call Hobbes]
Calvin is trying to call Hobbes but he doesn’t know if the number he’s calling is his home
phone or his work phone. The probability that Hobbes answers at his home phone is p
,
0
whereas the probability that Hobbes answers at his work phone is p
, with 0 < p
< p
< 1.
1
0
1
Calvin calls the number he has once a day until Hobbes finally picks up. Let X denote
the number of times Calvin has to call up until Hobbes finally answers (the answered call
included in the count). Assume that the successive call attempts are independent trials of an
experiment, so that X
Geometric(p
), under hypothesis H
. Assume that 2π
= 3π
.
i
i
0
1
(a) Find the ML decision rule in this binary hypothesis testing. Assume ties are broken in
favor of H
.
1
Solution: The likelihood ratio in this case is given by
k 1
P X = k H
p
(1
p
)
1
1
1
Λ(k) =
=
k 1
P X = k H
p
(1
p
)
0
0
0
3
ADVERTISEMENT
0 votes
Related Articles
Related forms
Mcas Math Test Worksheet With Answer Key - 7th Grade, Massachusetts Department Of Elementary And Secondary Education, 2013
Education
Taylor And Maclaurin Powers Series Worksheet With Answers - Math 141e Calculus And Analytic Geometry With Engineering Applications, Ii, Pennsylvania State University
Education
Related Categories
Parent category: Education