- Problem 3 In A Certain Lottery Tickets Are Distributed To People We Model The Prize Winnings For Each Ticket As Follow 1 (95.27 KiB) Viewed 154 times
Problem 3 In a certain lottery, tickets are distributed to people. We model the prize winnings for each ticket as following the distribution Exponential(A). Each ticket's winnings is independent of the others. We observe that if the winnings are too low, the prize is never claimed. In other words, there is some unknown parameter T such that if a prize is less than T, it will not be collected. Given the data 2 = (21,22, , 2n) of prize winnings which are collected, we would like to estimate T and A. As a prior distribution for T, we use the uniform distribution on (0,MT), where My is a very large positive number. Similarly, as a prior distribution for A we use the uniform distribution on [O, MA). You can assume that at least one the of 2; <M. (a) (5 points) Compute the P(x|A, T), the probability of prize winnings x1,... In being collected given parameters T and A. Note that if a prize winning were less than T it would not be claimed, so each of the I, must be at least as large as T. (b) (5 points) Compute ÎMAP, the MAP estimate of T given the data 21, ... , In under the given prior. Namely, compute Î MAP = arg max P(11..., |T =t, A = ). (c) (5 points) Compute AMAP, the MAP estimate of A given the data 11, ... , lin. Use your estimated ÎMap from part (a) and compute Âmap = arg max P (11,..„In/T = ÎMAP, 1 = 1) Suppose that the lottery company reveals to us that the parameter A is 1. We would like to estimate T (d) (5 points) Compute Îlms, the least mean square estimator of T, i.e. the estimator Î that minimizes 00 1 ſ (1 - 1)*P(T\)dt = p(2),.,on ) ! (1-1)ºple. In/?)m(T)QT | P(, 7AT |dT ... 0 (e) (5 points) Will ÎLMS converge to T in probability as no? Explain your answer.