We are given a biased coin, where the (random) bias Θ is uniformly distributed on [0,1] . We toss this coin, until we see Heads for the first time. Our goal is to estimate Θ using our observation.

Question

1. Find the LMS estimator (ΘˆLMS) for Θ . Your answer should be a function of N , the number of tosses up to and including the toss at which Heads were observed. (Make sure to use upper case N when entering your answer.)

2. Find E[Θ/(ΘˆLMS)] .

Answer 1

1. 2/(N+2)

2. 1