We are given a biased coin , where the probability of heads is q. he 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 the coin. We flip it 5 times, and define the(observed) random variable N as the number of heads in the experiment. Throughout this problem, you may find the following formula useful: x^n(1-x)^kdx= n!k!/(n+k+1)!.

1.Given the observation N=3. Calculate the posterior distribution of the bias Q.
2.What is the LMS estimate of Q, given N=3.
3.What is the resulting conditional mean squared error of the LMS estimator, given N=3.

1. To calculate the posterior distribution of the bias Q given the observation N=3, we can use Bayes' theorem. The posterior distribution is given by:

P(Q|N=3) = P(N=3|Q) * P(Q) / P(N=3)

P(N=3|Q) is the likelihood, which represents the probability of observing N=3 given a certain value of Q. In this case, it is the probability of getting exactly 3 heads in 5 flips of a biased coin. This can be calculated using the binomial distribution as:

P(N=3|Q) = (5 choose 3) * Q^3 * (1-Q)^2

P(Q) is the prior distribution of Q. Since Q is uniformly distributed on the interval [0,1], the prior distribution is simply 1 for all values of Q within this interval.

P(N=3) is the marginal likelihood, which is the probability of observing N=3 averaged over all values of Q. It can be calculated by integrating the product of the likelihood and the prior over the entire range of Q:

P(N=3) = ∫[0,1] (5 choose 3) * Q^3 * (1-Q)^2 dQ

To calculate this integral, we can use the formula provided:

P(N=3) = (5! * 2!) / (5+2+1)!

Once we have the likelihood, prior, and marginal likelihood, we can use Bayes' theorem to calculate the posterior distribution. The posterior distribution is proportional to the product of the likelihood and the prior, divided by the marginal likelihood:

P(Q|N=3) = P(N=3|Q) * P(Q) / P(N=3)

2. The LMS (Least Mean Squares) estimate of Q given N=3 is the mean of the posterior distribution. Since the posterior distribution is the distribution of Q given N=3, its mean represents the estimated bias of the coin based on the observation.

LMS estimate of Q = E[Q|N=3]

3. The resulting conditional mean squared error of the LMS estimator, given N=3, can be calculated as the variance of the posterior distribution. The conditional mean squared error represents the expected square difference between the estimated bias and the true bias.

Conditional mean squared error = Var[Q|N=3]

To calculate the variance, we need the second moment of the posterior distribution, which can be calculated as:

E[Q^2|N=3] = ∫[0,1] Q^2 * P(Q|N=3) dQ

From this, we can calculate the variance:

Var[Q|N=3] = E[Q^2|N=3] - (LMS estimate of Q)^2