Assume X= (X,..., X,) is a sample from a N(0, o2) distribution, where and o2 are unknown. Let and o2 have the joint improper noninformative

Assume X= (X₁,..., X,) is a sample from a N(0, o2) distribution, where and o2 are unknown. Let and o2 have the joint improper noninformative prior density

(0, 02) = 0-21(0.0) (02)

(In Subsection 3.3.3 it was stated that a reasonable noninformative prior for

(0, 0) is (0, o) = 010.0)(σ). This transforms into the above prior for (0, o²).)

(a) Show that the (formal) posterior density of and o2 given x is

π(θ, σ²x) = π(θ|σ², χ) π2(σ²x), where π,(θ]σ², x) isa.N(χ, σ²/n) density and π(σ²x) is an G((n-1)/2,

[3 Σ (χ,- x)2]-¹) density.

(b) Show that the marginal posterior density of o2 given x is an 9((n-1)/2,

[ (x-x)2]') density.

(c) Show that the marginal posterior density of given x is a I(n-1, х,

Σ",(x,-x)2/n(n-1)) density.