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 Yi = 1 if person i in the sample

supports the policy, and Yi = 0 otherwise.

a) Assume Y1, . . . , Y100 are, conditional on , i.i.d. binary random variables

with expectation . Write down the joint distribution of Pr(Y1 =

y1, . . . , Y100 = y100| ) in a compact form. Also write down the form of

Pr(

P

Yi = y| ).

b) For the moment, suppose you believed that 2 { 0. 0, 0. 1, . . . , 0. 9, 1. 0} .

Given that the results of the survey were

P100

i=1 Yi = 57, compute

Pr(

P

Yi = 57| ) for each of these 11 values of and plot these probabilities

as a function of .

c) Now suppose you originally had no prior information to believe one of

these -values over another, and so Pr( = 0. 0) = Pr( = 0. 1) = =

Pr( = 0. 9) = Pr( = 1. 0). Use Bayes rule to compute p (|

Pn

i=1 Yi =

57) for each -value. Make a plot of this posterior distribution as a

function of .

d) Now suppose you allow to be any value in the interval [0, 1]. Using

the uniform prior density for , so that p ( ) = 1, plot the posterior

density p ( ) ~ Pr(

Pn

i=1 Yi = 57| ) as a function of .

e) As discussed in this chapter, the posterior distribution of is beta(1+

57, 1+100 57). Plot the posterior density as a function of . Discuss

the relationships among all of the plots you have made for this exercise.