9. Show that the Jeffreys prior based on the binomial likelihood f(x|) = n x x(1 )nx is given by the Beta(.5,

9. Show that the Jeffreys prior based on the binomial likelihood



f(x|θ) = n x



θx(1 − θ)n−x is given by the Beta(.5, .5) distribution.

10. Suppose that in the situation of Example 2.3 we adopt the noninformative prior p(θ, σ) = 1

σ , θ ∈ , σ> 0 (i.e., p(θ, σ2) = 1

σ2 , θ ∈ , σ2 > 0).

(a) Show that the marginal posterior for t = √n(θ − y¯)/s, where s 2 = n i=1(yi − y¯)2/(n − 1), is proportional to

[1 + t 2/(n − 1)]−n/2 , a Student’s t distribution with (n − 1) degrees of freedom.

(b) Show that the marginal posterior for σ2 is proportional to

(σ2)

−[(n−1)/2+1] exp '

− 1 2σ2

n i=1

(yi − y¯)

2

(

, an IG n−1 2 , 2 n i=1(yi−y¯)2 density.