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, for i=1,2,…,k. (Ti includes the toss that results in the first Heads.)
You may find the following integral useful: For any non-negative integers k and m,
∫_0^1q^k(1−q)^m dq = k!m!/(k+m+1)!
1. Find the PMF of T1. (Express your answer in terms of t using standard notation.)
For t=1,2,…,
pT1(t)=
2. Find the least mean squares (LMS) estimate of Q based on the observed value, t, of T1. (Express your answer in terms of t using standard notation.)
E[Q∣T1=t]=
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