2. Suppose that X is distributed as a PoissOn random variable with mean A > 0 and that A has a prior distribution with density 0.) = I'(q)_1A7_le_)', A > 0, where 7 > 0 is a given constant. (a) Find the posterior distribution of A, its mean and its mode, given one observation of X. (b) Suppose now that X1, . . . ,Xn are independent observations of X. (i) Suggest a reasonable estimator of A and describe what happens to it as n, > 00. (ii) Show that the posterior predictive distribution of a future observation of X has the mass function T($+Z?=1Xi +7) f(:2:|X1, . . . ,Xn) oc 53! (71+ 2):: ,x=0,1,2,.... Hence deduce that the mode of the above posterior predictive distribution is attained t {max{zr=1Xi+7n2,U} a + 1 l, where [a] denotes the smallest integer 2 u. n 8. When making a purchase of wine, a restaurant is given the choice of 3 barrels, A, B or C. The restaurant wishes to choose the best, and, if barrels are ranked in true order of quality, their loss function will be equal to the rank of the chosen barrel minus 1. The restaurant employs two expert wine tasters who independently sample the barrels and rank them in order of quality. Each taster guesses the correct order with probability 2/5, guesses each of the two orderings which di'er from the true by interchange of a single neighbouring pair with probability 1/ 5, and guesses each of the three other orderings with probability 1/15. The expert tasters nominate orders BAC and ACE. Using a prior distribution which gives probability 1/6 to each of the 6 possible true orders, nd the posterior distribution of the true order. Hence nd the Bayesian decision which tells the restaurant which barrel to choose. 11}. A bag contains 91 chips, among which 62 are painted green and 61 62 are red. It is known that [91 E {l}, 1,2, . . .} and 6'2 E {l}, 1}, but the exact values of I? = (191,192) are unknown. Note that there can be at most one green chip in the bag. Suppose that our prior belief specifies that 6'1 m Poisson (A) and 6'2 ~ Bernoulli (q), for some given A > D and q E [[1, 1}5 and that I91 and 32 are independent of each other. Every dag;r a chip is drawn randomly from the bag without replacement and its colour recorded. Thus, by day 71, we have accumulated a random sample of observations {3:1, 3:2. . . . , 3\"}, where m; = 1 if a green chip is drawn on dag,r i and 112,-, : 0 otherwise. (3.) Write down an expression for the prior joint mass function of 6 = {191, I92). (b) Show that at the end of day a, the posterior joint mass function is given by 1T{|91,92|331,...,:I:n) Til. ' 3.1: -_ lIJ: or L61 L 3211 m 1_ m for 6'1 = 11,11 + 1, . . . and 6'; = U, 1. In the above expression an}r term of the form 0\" is interpreted as equal to 1. [-2) Suppose that there is a green chip found among the 11 chips drawn on the rst 71 days. Show that 1r{91,0|a:1,...,:tn) : 0, 6'1 =n,n+1,.... Hence deduce that the posterior odds for the existence of a green chip in the bag [:92 = 1) is P(62 = \"$1,. ...'$n) = 00 P192 2 Glxls' \"rmn) (d) Consider the case where a = ... = In = 0. Let W denote a Poisson random variable with mean 1. (i) Show that the posterior odds for the existence of a green chip in the bag is P(02 = 1|0, . . ., 0) P(02 = 0|0, . . .,0) (1 - nE[W -|W zn]). (ii) Show that IE[0 10, . . . .0 = EWW 2n - qn 1 - qn E[W '(W > n]11. Story of Cinderella (episode 2) continuation of Assignment 1/628... [Recall that in episode 1, Cinderella danced with the Prince at his palace. She had come on a horsedrawn coach, wearing a pair of glass slippers, but left the palace just before midnight...] The Prince's exwife was scheduled to return after midnight to the palace. The Prince knew that he had to give up half of his palace to his eat-wife once she arrived. The Prince's palace was worth 2'30 million dollars. The Prince's true reason for throwing the royal ball was to choose a rich wife among the guests to help him recover his anticipated loss. The Prince found that Cinderella was probably the only lady who looked rich enough to help him. Unfortunately, she had left the palace in a rush at midnight, leaving only a glass slipper en the palace steps. The Prince stared at the slipper, wondering whether he should bring the slipper back to Cinderella in person. If the Prince chooses to stay put at his palace, he will gain a slipper, which is worth 100 dollars, but half of his palace, which is worth 10!] million dollars, will be lost to his err-wife. If the Prince chooses to bring the slipper back to Cinderella, he will lose the whole palace, for the palace cannot be defended in his absence. 0n the other hand, his sincerity will touch Cinderella so deeply that she will marry him immediately. In this case Cinderella's wealth, which is worth an unknown amount of :9 dollars, will all go to him. To help him gauge Cinderella's wealth, the Prince made observations about the type of slippers (Y1) worn by Cinderella and her means of tran8port (Y2) to the palace. It is known that (Y1, Y2} has the following mass function: Y; \\ Y1 1 {leather} 2 {glass} 3 (diamond) 1 (walk) 0.5{1 + 15')'2 0.5(1 + 3)'1 [1.5 6(1 + Elf:2 2 (coach) 0.53(1+ Ell2 {153(1 + fill1 [1592(1 + Ell2 (a) Let so be the action of staying put at the palace, and m be the action of bringing the slipper back to Cinderella. Set up the loss function Lug\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