8. Suppose that f(x|, ) is normal with mean and standard deviation . (a) Suppose that is known, and show that the Jeffreys

8. Suppose that f(x|θ, σ) is normal with mean θ and standard deviation σ.

(a) Suppose that σ is known, and show that the Jeffreys prior (2.12) for

θ is given by p(θ)=1, θ ∈ .

(b) Next, suppose that θ is known, and show that the Jeffreys prior for

σ is given by p(σ) = 1

σ , σ> 0 .

(c) Finally, assume that θ and σ are both unknown, and show that the bivariate Jeffreys prior (2.15) is given by p(θ, σ) = 1

σ2 , θ ∈ , σ> 0 , a slightly different form than that obtained by simply multiplying the two individual Jeffreys priors obtained above.