4 solution.pdf - STAT 24400 Problem Set 4 Solutions Total...

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STAT 24400 Problem Set 4 Solutions Total points: 100 1. [20pts] Evaluation Normal distribution probabilities Note that the table is only for standard normal distribution. X N ( - 4 , 16) ) Y ( X + 4) / 4 N (0 , 1) . (a) P ( X > 2) = P ( Y > 1 . 5) = 1 - Φ (1 . 5) = 0 . 0668 (b) P (0 < X < 4) = P (1 < Y < 2) = Φ (2) - Φ (1) = 0 . 1359 (c) P ( | X + 3 | 3) = P ( Y  - 0 . 5 or Y 1) = 1 - Φ (1) + Φ ( - 0 . 5) = 0 . 4672 (d) P ( X 0 or X 3) = P ( Y 1) + P ( Y 1 . 75) = 1 - Φ (1 . 75) + Φ (1) = 0 . 8814 Grading Scheme : 5 pts for each part. 4/5 credit for correct setup in terms of normal cdf. 2. [20pts] Bayesian Inference for Psychics Let A denote the event that the guesser is psychic, meaning that she guesses correctly with probability 1 / 2. Then A c is the event that she is not psychic, meaning that she guesses correctly with probability 1 / 5. Let B be the event that she correctly guesses 3 out of the 5 cards. We also need a prior probability that the guesser is psychic, P ( A ). We’ll call this probability p . Then our posterior probability is P ( A | B ) = P ( B | A ) P ( A ) P ( B | A ) P ( A ) + P ( B | A c ) P ( A c ) = ( 5 3 ) (1 / 2) 3 (1 / 2) 2 p ( 5 3 ) (1 / 2) 3 (1 / 2) 2 ( p ) + ( 5 3 ) (1 / 5) 3 (4 / 5) 2 (1 - p ) = 5
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