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Problem 3 [15 marks] The speed limit on Gunghalin Drive in Canberra between Barton Highway and the Glenloch interchange is 90km/h. Ben frequently drives on Gunghalin Drive and typically drives at a constant speed of 90km/h (that is, at the speed limit) on this section of road. One day, he passes 3 cars and gets passed by 17 cars on this section of road. Suppose that car speeds on this section of road are normally distributed with unknown mean p. and known standard deviation 0 = 4.5. Let s = 3 denote the number of cars that Ben overtakes and let their unobserved car speeds be y1,y2,y3. Let t = 17 denote the number of cars that overtake Ben and let their unobserved car speeds be y4, y5, ..... ygo. (a) [3 marks] Assign the unknown mean p. a at prior density. Write down the mathematical expression for the posterior density of p. (that is, p(,u.|a, s, t, y)) up to a proportionality constant. Hint: the actual car speeds ya: (2' = 1, ...,20) are not observed, but if Ben passes say Car A, then we know that the speed of Car A must be less than 90km/ hr. Similarly if Car B passes Ben, then we know that the speed of Car B must be greater than 90km/ hr. (b) [2 marks] Plot the posterior density of a. (c) [1 mark] Using the density found in part (b), provide a 95% interval estimate for the average speed at which cars travel along this section of Gunghalin Drive between Barton Highway and the Glenloch interchange. (d) [1 mark] Estimate the probability that the average speed of the cars on this section of Gunghalin Drive exceeds the 90km/ h speed limit. (e) [2 marks] Now let's assume a is unknown. Assume the non-informative joint prior distribution p(,u,02) 0c (a2)'1. Derive the joint posterior distribution of (,u, 02) up to a proportionality constant. (f) [3 marks] Create a contour plot of the joint posterior density of ,u and (72. (g) [3 marks] Using the joint density found in part (f ), provide a 95% interval estimate for the average speed at which cars travel along this section of Gunghalin Drive between Barton Highway and the Glenloch interchange. How does your answer compare to your previous answer in part (c)