13. Chocolate bars of nominal weight 50g are produced in a single production run. Records show that the actual weight of each chocolate bar deviates from the nominal weight by a random error normally distributed with mean zero and standard deviation or (in grams). It may he assumed that 0 is xed within each run but varies from run to run randomly such that 1 f 02 has the prior densityr function 11T{s)oc s'le'an, s > U. A sample of n {23 2) chocolate bars is taken from a single production run and their weights are recorded as W1, W2? . . . , W {in g). (a) Show that the posterior density function of (:2, given W1, . . . T Wm is 1.1 I'VE502 1 n(t|W1,...,Wn) oc Elma"? exp { W} , t> CI. (b) {0) Hence show that the posterior mode and mean of 02 are given by 1 n 1 n n + 3 {2092: - 50]2 +1} and n _1{Z[VV, 50}2 +1} , respectively. i=1 i=1 at Hint: For p. E (oo,oo) and o > U. the density function of N(p,o2) is its) = oweWes { (IQ(,5? GIG * Hint: f feeWas = bl_ar(a 1) for any a > 1. b > o. 0 }, oo