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# homework6 - IE 300 GE 331 Spring 2009 Homework#6 Problem 1...

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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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