Problem 2 [15 marks] Let Yi, Yz, .... Yn be independent and identically distributed (IID) random variables with common density function friA.B(yla, b) = ae-aby for y > 0. Sample data y1, . .., In are assumed to be observations of Yi, . .. . Yn. (a) Consider an exponential prior pA(a) = e-"I(a > 0). For a fixed b > 0, find the posterior distribution PA.......B(aly1, ....Un, b). Identify the name and parameter of this distribution. [3 marks] (b) For the posterior distribution in (a), for b = 0.5, n = 10. Zig, yi = 10, compute the posterior expectation and posterior variance of A. [3 marks] (c) Based on (b), construct a 95% Bayesian credible interval for A. How do you inter- pret this interval? Explain how is the interpretation different from the one for a 95% confidence interval constructed from a frequentest ML approach. [4 marks] (d) Now we assume A and B are both random variables. Consider priors pA(a) = e-"I(a > 0) and pa(b) o be I(b > 0). Use Gibbs sampling to get 95% credible intervals for A and B. Assume n = 10 and > > = 10. Provide code and steps to derive the posterior distributions required by your algorithm. [5 marks]