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Problem 1 [10 marks] How many ambulance vehicles are there in Canberra? Suppose I know there cannot be more than 60 ambulance vehicles in Canberra. Whilst driving around last week I observed six ambulances numbered 13, 14, 21, 26, 37 and 38. So the sample size is n = 6. I assume that ambulances in Canberra are numbered from 1 to N, and that I am equally likely to observe any numbered ambulance at any time. I also assume observations are independent. To solve this problem, suppose one takes independent observations y1, ..., yn from a discrete uniform distribution on the set {1, 2, ..., N}, where the upper bound N is unknown. Suppose one places a uniform discrete prior for N on the values 1, ..., B, where B is known. (a) [3 marks] Derive the posterior distribution of N up to a proportionality constant. Be sure to specify the bounds on the parameter space of N in your posterior distribution. (b) [3 marks] Compute posterior probabilities of N over a grid of values. (c) [2 marks] Compute the posterior mean and posterior standard deviation of N. (d) [2 marks] Find the posterior probability that there are more than 50 ambulance vehicles in Canberra