This exercise will prove the assertions in Example 7.2.16, and more. Let X1,..., Xn be a random sample from a n(θ, σ2) population, and suppose that the prior distribution on θ is n(μ, τ2). Here we assume that σ2, μ, and τ2 are all known.
a. Find the joint pdf of X and θ.
b. Show that m(|σ2,μ, τ2), the marginal distribution of , is n(μ, (σ2/n) + τ2).
c. Show that π(θ|, σ2, μ, τ2), the posterior distribution of θ, is normal with mean and variance given by (7.2.10).