3.1 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 vari- ables with expectation ?. Write down the joint distribution of Pr(Y1 = y1, .P. . , Y100 = y100|?) in a compact form. Also write down the form of Pr( Yi=y|?). b) For the moment, suppose you believed that ?P2 {0.0, 0.1, . . . , 0.9, 1.0}. GivPen that the results of the survey were 100 Yi = 57, compute i=1 Pr( Yi = 57|?) for each of these 11 values of ? and plot these prob- abilities 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.P1) = = Pr(? = 0.9) = Pr(? = 1.0). Use Bayes' rule to compute p(?| ni=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 Pdensity for ?, so that p(?) = 1, plot the posterior density p(?) ? Pr( ni=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.

3.1 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, = 1 if person 3' in the sample supports the policy, and Y,- = 0 otherwise. a) Assume Y1, . . . 13/100 are, conditional on 6, i.i.d. binary random vari- ables with expectation 6. Write down the joint distribution of Pr(Y1 : yl, . . . , Y100 : ylool) in a compact form. Also write down the form of m: V. = 946). b) For the moment, suppose you believed that 9 E {0.0, 0.1, . . . , 0.9, 1.0}. Given that the results of the survey were 231:? Y,- = 57, compute Pr(Z Y,- : 57l6') for each of these 11 values of 9 and plot these prob- abilities as a function of 19. c) Now suppose you originally had no prior information to believe one of these G-values over another, and so Pr(6' : 0.0) = Pr(6' = 0.1) : ... : Pr(9 : 0.9) : Pr(9 : 1.0). Use Bayes\" rule to compute p(6|Z:;11'} : 57) for each Q-value. Make a plot of this posterior distribution as a function of 6. (:1) Now suppose you allow 6' to be any value in the interval [0, 1]. Using the uniform prior density for 6, so that p(6) : 1, plot the posterior density p(9) >