Suppose that, as in Example E of Section 3.5.2, your prior opinion that the coin will la...

Suppose that, as in Example E of Section 3.5.2, your prior opinion that the coin will land with heads up is represented by a uniform density on [0, 1]. You now spin the coin repeatedly and record the number of times, N, until a heads comes up. So if heads comes up on the first spin, N = 1, etc.

a. Find the posterior density of given N.

b. Do this with a newly minted penny and graph the posterior density.

Reference

Bayesian Inference A freshly minted coin has a certain probability of coming up heads if it is spun on its edge, but that probability is not necessarily equal to 1 2. Now suppose it is spun n times and comes up heads X times. What has been learned about the chance the coin comes up heads?We will go through a Bayesian treatment of this problem. Let , we might represent our state of knowledge by a uniform density on [0, 1]:

We will see how observing X changes our knowledge about , transforming the prior distribution into a “posterior” distribution.

Given a value θ, X follows a binomial distribution with n trials and probability of success θ: