1. Sample data, y, on the number of Bernoulli "trials"required forn "successes" is assumed to be a value of Y ~ NegBin (n, #). The objective here is to estimaten via Bayes Rule. The random variable II is introduced to represent uncertainty about. The prior distribution for II is taken to be Beta (a, b)- (a) [3 marks] Show that the Bayesian posterior distribution of II given data/ is a beta distribution with its parameters taking valuesa + n and b ty -n, respectively. (b) [2 marks] Is the beta distribution a conjugate prior here? Why or why not? (c) Patients with high cholesterol entering a medical study are given an experimental drug. Each patient has his/her cholesterol status assessed as still "elevated" or not after one week of treatment. Recruitment of patients stops when = 64 patients are assessed to have elevated cholesterol; this requiresy = 296 patients to enter the study. It is assumed that y is the value of Y ~ NegBin (n, #), where a is the probability of elevated cholesterol. The prior for II is Unif (0, 1). i. [1 mark] What is the posterior distribution of II? ii. [1 mark] Numerically, what is the posterior expectation of