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The profit per client-day made by a privately owned health centre depends on the variable costs involved. Variable costs, over which the owner of the health centre has no control, take one of turee levels 61 = high, 8, = most likely, 83 = low. The owner has to decide at what level to set the number of client-days for the coming year. Client- days can be either dj - 16, d2 - 13.4 or d3 - 10 (each in 000s). The profit (() per client-day is as follows: d 85 95 110 dz 105 115 130 125 135 150 (i) Determine the Bayes criterion solution based on the annual profits, given the probability distribution p(0, ) =01, p(62) =06, p(63) =03. [3] (ii) Determine both the minimax solution and the maximax solution to this problem. [2] [Total 5]The number of claims per year from individual policies in a portfolio is believed to follow a Poisson(1) distribution. A Gamma (5,2) distribution is chosen as a prior distribution for 2. A random sample of 10 policies is observed over the last year, and the numbers of claims are found to be as follows: 4, 1, 0, 0, 2, 0, 0, 1, 3, 1 (i) Derive the posterior distribution for 1 131 (ii) Hence, determine the Bayesian estimate for 1 : (a) under squared error loss (b) under zero-one loss (c) under absolute error loss. [5] [Total 8]