6. Biased coin

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Problem 5. Biased coin
5.0 points possible (graded, results hidden)
We are given a biased coin, where the probability of Heads is q. The bias q
is itself the realization of a random variable Q which is uniformly distributed on the interval [0,1]. We want to estimate the bias of this coin. We flip it
5 times, and define the (observed) random variable N as the number of Heads in this experiment.

Given the observation N=3, calculate the posterior distribution of the bias
Q. That is, find the conditional distribution of Q, given N=3.

For 0≤q≤1,

Question 1 --> fQ|N(q∣N=3)= ?

Question 2 --> What is the LMS estimate of Q, given N=3?

Question 3 --> What is the resulting conditional mean squared error of the LMS estimator, given N=3?

Answer 1

Answer 1: fQ|N(q∣N=3)= (5q^3)(1-q)^2
Answer 2: The LMS estimate of Q, given N=3 is q=3/5.
Answer 3: The resulting conditional mean squared error of the LMS estimator, given N=3 is 1/25.