Consider a normal hierarchical model in that observations x1, . . . , xn are a random sample from a normal distribution with mean and variance 1/. Assume further that the prior for is a normal distribution with mean 0 and variance /, and the prior for is a gamma distribution with shape a and rate b. Schematically, xi N(, 1 ), i = 1, . . . , n, N(0, ), gamma (a, b). Assume further that the values of 0, , a, and b are known and answer the following. (a) Write down the joint distribution of x1, . . . , xn, and . (b) Find the conditional distribution of given x1, . . . , xn and . Simplify expressions as much as you can. 2 (c) Find the conditional distribution of given x1, . . . , xn and . Again, simplify expressions as much as you can. (d) Describe one full iteration of the Gibbs sampler to draw from the posterior density of and given data x1, . . . , xn