4. [20 marks] Suppose that the Metropolitan Bus Company is considering extending a special commuter service to an outlying suburb and that there are 200 commuters in this suburb who might use such a service. According to the plan, a bus will transport commuters from the suburb to the CBD at a specified time each morning and a return trip will be made each evening. A fare of $5 is to be charged for each one-way trip and no special round trip fare is contemplated. The total cost of the new service would be $270 per day. A decision is to be made as to whether or not to extend this service. Now suppose that the bus company has information about experiences with special commuter bus services in other cities, indicating that either a fraction of 8, = 0.1, 8, = 0.15. or 0x = 0.2 of the commuters will take up the bus service with a probability of P: = 0.6. P2 = 0.25 and ps = 0.15, respectively. Let az = 1 denote the action to introduce bus service", and a2 = 2 "not introduce bus service". (a) [2 marks] Describe the actions, states of nature and enter the monetary conse- quences in S's per day, a negative value corresponds to a profit, a positive value to a loss) of each decision as losses in a loss matrix. (b) [3 marks Calculate the prior expected losses for decisions a, and a2. State the Bayes decision. Would this change if you used a uniform prior instead? (c) (1 marks] What is the minimax decision? (d) [1 marks] Now suppose, that a sample of n = 20 commuters is taken and the number X observed, who indicate that they would use the new service. How many possible decision rules d(X) are there? (e) 15 marks] Consider the following three decision rules: dir) = di, if I > 2 and an, if 152 if > 3 d2() and d3() 21, if > 4 22. if