We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0, 1]. We assume that conditioned on Q -q, all coin tosses are independent. Let Ti be the number of tosses of the ith coin until that coin results in Heads for the first time, fori- 1,2,..., k. (Tt includes the toss that results in the first Heads.) You may find the following integral useful: For any non-negative integers k and m k!m! e* (1-q)" dq = (k + m + 1)! 1. Find the PMF of T1.(Express your answer in terms of t using standard notation.) For t = 1, 2, . . .' PT (t) 2. Find the least mean squares (LMS) estimate of Q based on the observed value, t, of Ti.(Express your answer in terms of t using standard notation.) 3. We flip each of the k coins until they result in Heads for the first time. Compute the maximum a posteriori (MAP) estimate of Q given the number of tosses needed, Tİ-ti , . . . , T = tk, for each coin. Choose the correct expression for q k -1 k +1 O none of the above

Note:
Complex mathematical expressions are in the question