Suppose that X1, X2,..., X, are a random sample from a Poisson(1) distribution. Also, let X have a Gamma(a, ) prior distribution, the conjugate family for the Poisson. Recall the posterior distribution for given y = Li-l li, derived for Assignment 2. Using those results, consider a Bayesian test of Ho : 1 lo (a) Calculate expressions for the posterior probabilities of H, and H. (Simply describe the integrals/densities of interest without trying to simplify the expression.) (b) Consider a test that rejects Ho :) 3 if and only if P15 3\y) 3 y). Assume that a = 5/2 and B 2 for the prior distribution. Suppose that you observe a sample such that y = 25 for n = 10. Calculate the posterior probabilities for H, and H and make a decision to reject (or not reject) H.. (You will need to use the pgamma(?, shape=?, scale=?) function in R to evaluate the posterior probabilities) Suppose that X1, X2,..., X, are a random sample from a Poisson(1) distribution. Also, let X have a Gamma(a, ) prior distribution, the conjugate family for the Poisson. Recall the posterior distribution for given y = Li-l li, derived for Assignment 2. Using those results, consider a Bayesian test of Ho : 1 lo (a) Calculate expressions for the posterior probabilities of H, and H. (Simply describe the integrals/densities of interest without trying to simplify the expression.) (b) Consider a test that rejects Ho :) 3 if and only if P15 3\y) 3 y). Assume that a = 5/2 and B 2 for the prior distribution. Suppose that you observe a sample such that y = 25 for n = 10. Calculate the posterior probabilities for H, and H and make a decision to reject (or not reject) H.. (You will need to use the pgamma(?, shape=?, scale=?) function in R to evaluate the posterior probabilities)