You are going to conduct a survey of the voters in the city you live in. They are being asked whether or not the city should build a new convention facility.

You believe that most of the voters will disapprove the proposal because it may lead to increased property taxes for residents. As a resident of the city, you have been hearing discussion about this proposal, and most people have voiced disapproval. You think that only about 35% of the voters will support this proposal, so you decide that a beta (7, 13) summarizes your prior belief.

However, you have a nagging doubt that the group of people you have heard voicing their opinions is representative of the city voters. Because of this, you decide to use a mixture prior:

g(π|i) =
g0(π) if i = 0 g1(π) if i = 1 , where g0(π) is the beta (7, 13) density, and g1(π) is the beta (1, 1) (uniform)

density. The prior probability P(I = 0) = .95. You take a random sample of n = 200 registered voters who live in the city. Of these, y = 10 support the proposal.

(a) Calculate the posterior distribution of π when g0(π) is the prior.

(b) Calculate the posterior distribution of π when g1(π) is the prior.

(c) Calculate the posterior probability P(I = 0|Y ).

(d) Calculate the marginal posterior g(π|Y ).