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 446 Bayesian Statistical Inference Chap. 8

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.