Sample survey: Suppose we are going to sample 100 individuals from a county (of size much larger than 100) and ask each sampled person whether they support policy Z or not. Let Y_i = 1 if person i in the sample supports the policy, and Y_i = 0 otherwise. a) Assume Y_1, ..., Y_100 are, conditional on theta, i. i.d. binary random variables with expectation theta. Write down the joint distribution of Pr(Y_1 = y_1, ..., Y_100 = y_100|theta) in a compact form. Also write down the form of Pr (sigma Y_i = y| theta).
b) For the moment, suppose you believed that theta elementof {0.0, 0.1, ..., 0.9, 1.0}. Given that the results of the survey were sigma^100 _i = 1 Y_i = 57, compute Pr (sigma Y_i = 57|theta) for each of these 11 values of theta and plot these probabilities as a function of theta.
c) Now suppose you originally had no prior information to believe one of these theta-values over another, and so Pr(theta = 0.0) = Pr(theta = 0.1) =... = Pr(theta = 0.9) = Pr(theta = 1.0). Use Bayes' rule to compute p{theta| sigma^n _i = 1 Y_i = 57) for each theta-value. Make a plot of this posterior distribution as a function of theta.
d) Now suppose you allow theta to be any value in the interval [0, 1]. Using the uniform prior density for theta, so that p(theta) = 1, plot the posterior density p(theta) times Pr(sigma^n _i = 1 Y_i = 57|theta) as a function of theta.
e) As discussed in this chapter, the posterior distribution of theta is beta(1 + 57, 1 + 100 - 57). Plot the posterior density as a function of theta. Discuss the relationships among all of the plots you have made for this exercise.