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.

Question

1) 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,

2) What is the LMS estimate of Q, given N=3?

Answer 1

1.140*y^3*(1-y)^3
2.0.5
3. What is the resulting conditional mean squared error of the LMS estimator, given N=3?
ans. 1/36

Answer 2

any hint, how did you do the calculation?

Answer 3

This is the case for flipping 6 times.

Answer 4

Hint: Use a normalization constant for the Bayes theorem