09 02 2023 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Beraha 09.02.2023 Properly justify all your answers. Use the indicator function to denote the support of a distr. . Exercise 1 It is well-known that FAIL/PASS outcome of theStatistics course exam for each student is random. Hence, all the students attend all the exam sessions until they pass the exam. Assume that there is an infinite sequence of the exam occurrences and that, for each student, the outcome of the occurrences are (conditional) independent with the same probability p of passing the exam. For a student i, denote by Xi the number of exam occurrences to give in order to pass the exam. In particular, Xi = 1 if the student get a PASS mark at the first shot. 1. Verify that the conditional distribution of any Xi, given p ∈ (0, 1), is the following discrete density P (Xi = k|p) = p (1 − p)k−1 k = 1, 2, 3, . . . , (1) and that E[ Xi | p] = 1/p. Find also the analytic expression of P (Xi > k |p) for a fixed k. (Hint: Remember that P+∞ j=0 qj = 1 1 − q if 0 < q < 1. This formula is useful to compute both E[Xi | p] and P (Xi > k |p). ) We have only access to partial data from the last academic year: of the 60 students enrolled in the course, 15 of them passed the exam on the first occurrence, 12 of them on the second occurrence, and 18 on the third occurrence. We do not have any information about the remaining students but that they took more than 3 tries to pass the exam. Assume that, conditioning to p, all the students’s outcome are independent, with the same success probability p. 2. As the joint likelihood for X1, . . . X60, consider the information that you have, i.e., that X1, . . . , X45 are totally observed, while X46, . . . , X60 are partially observed as described before. Compute the likelihood for the available data.…