Let X1, X2, . . . , Xn be a random sample from a gamma distribution with known α and with θ = 1/τ. Say τ has a prior pdf that is gamma with parameters α0 and θ0, so that the prior mean is α0θ0.
(a) Find the posterior pdf of τ, given that X1 = x1, X2 = x2, . . . , Xn = xn.
(b) Find the mean of the posterior distribution found in part (a), and write it as a function of the sample mean and α0θ0.
(c) Explain how you would find a 95% interval estimate of τ if n = 10, α = 3, α0 = 10, and θ0 = 2.