Question

Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and (1- Yn) are the weights. Thus, the posterior mean is the weighted average of the sample average X and the prior mean lo. b) Show that the weight on the sample average goes to one as the sample size goes to infinity. Symbolically, show, lim y,-1. This indicates that the prior distribution has little impact for large sample sizes. Suppose that two statisticians A and B assign to θ the following different prior c) pdfs to θ: ξ(e) 90<b<l and ξ,0)-4830<θ<1. 2 ote that both prior distributions are Beta distributions. If a sample of n 1000 yields 710 successes, then give the Bayesian estimates of e for each statistician under squared-error loss. Bayesian,A Bayesian,

Answer 1