An experiment was performed to estimate the effect of beta-blockers on the one-year mortality of individuals with advanced cardiac disease (ACD). A group of n = 700 randomly selected individuals with ACD were treated with beta-blockers. Y = 22 of these individuals died within one year. Let 0 be the probability of death within one year for individuals with ACD who were treated with beta-blockers. What is the posterior density of the probability of death within one year, p(011), for individuals with ACD taking beta-blockers? You must show the derivation for full points and provide the name and parameters of the distribution if it has a named form. You may assume that the number of deaths follows a binomial distribution and that you have no prior information about the death probability (i.e., you should assume that 01 Unif(0, 1)). Draw 1000 samples from the posterior distribution and provide a histogram of these samples. ~ A group of n = 674 randomly selected individuals with ACD were selected and were not treated with beta-blockers. These individuals were chosen independently of the individuals chosen in part (a). Yo = 39 of these individuals died within one year. Let 0 o denote the probability of death within one year for individuals not treated with beta-blockers. What is the posterior distribution of the probability of death within one year, p(00|yo), for individuals with ACD not taking beta-blockers? You do not need to perform the derivation again if you can generalize your result from part (a). Again, you may assume that the number deaths follows a binomial distribution and that you have no prior information about the death probability. Draw 1000 samples from the posterior distribution of 0 and provide a histogram of these samples. For each of your 1000 samples from the posterior distribution of 01 and the posterior distribution of 0, say 0] and [] for k = 1, , 1000, compute -0[6]). It can be shown that these samples are from the posterior distribution of the difference in the one-year death probabilities of individuals with ACD taking beta- blockers versus those not taking beta-blockers. That is, the samples are from the posterior distribution, p(01-001, yo). Use the samples from this distribution to approximate E[01 - 001, 0], Var[01 001, 90], and the posterior probability that the one-year death rate of ACD individuals taking beta- blockers is less than the one-year probability of death for individuals not taking beta blockers.