Let X1, . .. . . Xn be a random sample from a n (θ, σ2) population, and suppose that the prior distribution on θ is n(μ, T2) Here we assume that σ2, μ, and T2 are all known.
(a) Find the joint pdf of X and θ.
(b) Show that m(x/σ2, μ, T2) , the marginal distribution of X, is n(μ, (σ2/n +T2)
(c) Show that π (θ|x, σ2 , μ, T2) , the posterior distributions of θ, is normal with mean and variance given by (7.2.10)
