Exercise 5.4 Show that the normal model with unknown mean and variance belongs to the two-parameter exponential family. Use expression (5.3) to derive the natural conjugate prior of the normal distribution. Finally, show that the extended conjugate family is needed p(θ | α,β, γ) = k(α,β, γ) exp  c(θ) Tα + βd(θ) + γe(θ)  , (5.39) where α, β are hyperparameters as in expression (5.3) and γ an extra set of parameters, to yield the normal-inverse-gamma prior (O’Hagan and Forster 2004, pp. 140–141).