Students in a probability class take a multiple-choice

Chapter , Problem 4

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Students in a probability class take a multiple-choice test with 10 questions and 3 choices per question. A student who knows the answer to a question will answer it correctly, while a student that does not will guess the answer with probability of success 1/3. Each student is equally likely to belong to one of three categories i = 1, 2, 3: those who know the answer to each question with corresponding probabilities Oi, where 01 = 0.3, 02 = 0.7, and 03 = 0.95 (independent of other questions). Suppose that a randomly chosen student answers k questions correctly. (a) For each possible value of k, derive the MAP estimate of the category that this student belongs to. (b) Let M be the number of questions that the student knows how to answer. Derive the posterior PMF, and the MAP and LMS estimates of M given that the student answered correctly 5 questions.