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

{ c e

-O.04x

!Tle (x I e = 1) = 0

1 ,

if 8 = 1 (problem is difficult), and is

if 5 x 60,

otherwise,

{ c eO.

16x if 5 x 60, !Tle (x I e = 2) = 0

2 '

otherwise,

if e = 2 (problem is not difficult) , where Cl and C2 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?

(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 solution

times are 10, 25, 15, and 35 minutes. On the basis of the five observations, which

hypothesis will she now accept, and what will be the probability of error?