Suppose we have an i.i.d. sample XiN(1,12), i = 1, . . . , n, and another independent i.i.d. sample YjN(2,22), j = 1, . . . , m. The parameters k,k2, k = 1, 2 are all unknown.

(a)Suppose we adopt a Bayesian approach and use priorsp(k,k2) _k² for k = 1, 2 independently. Show that the marginal posterior distributions of 12and22 can be expressed as (n1)sX2/12Datan12and(m1)sY2/22Datam12 independently. (You may find the following identity usefulni=1(xic)2=ni=1(xix)2+n(xc)2

(b) In the setting of part (a) show that a Bayesian 95 percent credible interval for 12/22can be based onsX2/sY2 and how is this credible interval different from the confidence interval?