iid Prove Proposition 2.9. That is, suppose y; ~ N(u, o') with / and o unknown, and i = 1, ...,n. If the (improper) prior density for (u, o?) is p(u, o?) oc 1/02, prove that 1. plot, y ~ N(y. 02/ n) 2. olly ~ inverse-Gamma 1-1 where S = _ (); -y)2. Consult the definitions of the normal and inverse-Gamma densities provided in Appendix B.iid Proposition 2.9 If yi ~ N(u, 02) and p(u, 02) ox 1/02 then p(u, a? ly) o (03)7-1 exp 1 202 (S + n (5 - 1) 2 ) where S = Ell_ ()i - $)2 and )=n-')_, yi. Equivalently, uo',y - N(,o') o2ly ~ inverse-Gamma n - 2 H - y ow In-1 S/ ((n - 1)n)