You are going to conduct a survey of the students in your university to find out whether they read the student newspaper regularly. Based on your friends opinions, you think that a strong majority of the students do read the paper regularly. However, you are not sure your friends are representative sample of students. 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 (20, 5) 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 = 100 students. Of these, y = 41 say they read the student newspaper regularly.

(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 )