Topic: Bayesian inference. Estimating the parameter of a geometric r.v.

Problem 7. Estimating the parameter of a geometric r.v. 3 points possible (graded) We have * coins. The probability of Heads is the same for each coin and is the realized value q of a random variable @ that is uniformly distributed on [0, 1). We assume that conditioned on @ = q, all coin tosses are independent. Let 7, be the number of tosses of the ?" coin until that coin results in Heads for the first time, for ? = 1, 2, ..., k. (It includes the toss that results in the first Heads.) You may find the following integral useful: For any non-negative integers k and in, kim! 1. Find the PMF of 71 - (Express your answer in terms of t using standard notation.) Fort = 1, 2, . . ., Pn (t) = 2. Find the least mean squares (LMS) estimate of @ based on the observed value, t, of 71. (Express your answer in terms of { using standard notation.) E [Q | T = +] = 3. We flip each of the k coins until they result in Heads for the first time. Compute the maximum a posteriori (MAP) estimate q of @ given the number of tosses needed, Ti = t1, . . ., Ik = th, for each coin. Choose the correct expression for q. O 1 = k - 1 O k 4 = Oi 1 = k +1 Ziti Onone of the above