IE 300 / GE 331 Spring 2009
Homework #6
Problem 1.
Professor May B. Hard, who has a tendency to give difficult problems
in probability quizzes, is concerned about one of the problems she has prepared for
an upcoming quiz. She therefore asks her TA to solve the problem and record the
solution time.
May’s prior probability that the problem is difficult is 0.3, and she
knows from experience that the conditional PDF of her TA’s solution time
X
, in
minutes, is
f
T

Θ
(
x

Θ = 1) =
(
c
1
e

0
.
04
x
,
if 5
≤
x
≤
60
0
,
otherwise
,
If Θ = 1 (Problem is difficult), and is
f
T

Θ
(
x

Θ = 2) =
(
c
2
e

0
.
16
x
,
if 5
≤
x
≤
60
0
,
otherwise
,
If Θ = 2 (Problem is not difficult), where
c
1
and
c
2
are normalizing constants. She
uses the MAP rule to decide whether the problem is difficult.
(a) Given that the TA’s solution time was 20 minutes, which hypothesis will she
accept and what will be the probability of error? (Note: the problem is asking
about the probability of error given her decision and not for the MAP rule.)
(b) Not satisfied with the reliability of her decision, May asks four more TAs to
solve the problem. The TAs’ solution times are conditionally independent and
identically distributed with the solution time of the first TA. The recorded solu
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 Spring '09
 NegarKayavash
 Probability, Probability theory, TA

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