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The punctuality of trains has been investigated by considering a number of train journeys. In the sample, 60% of trains had a destination of Manchester, 20% Edinburgh and 20% Birmingham. The probabilities of a train arriving late in Manchester, Edinburgh or Birmingham are 30%, 20% and 25%, respectively. A late train is picked at random from the group under consideration. Calculate the probability that it terminated in Manchester. 2 A random variable X has a Poisson distribution with mean 1, which is initially assumed to have a chi-squared distribution with 4 degrees of freedom. Determine the posterior distribution of 1 after observing a single value x of the random variable X . 3 The number of claims in a week arising from a certain group of insurance policies has a Poisson distribution with mean # . Seven claims were incurred in the last week. The prior distribution of / is uniform on the integers 8, 10 and 12. (i) Determine the posterior distribution of # . (ii) Calculate the Bayesian estimate of # under squared error loss. 4 For the estimation of a population proportion p, a sample of n is taken and yields x successes. A suitable prior distribution for p is beta with parameters 4 and 4. (i) Show that the posterior distribution of p given x is beta and specify its parameters. [2] (ii) Given that 11 successes are observed in a sample of size 25, calculate the Bayesian estimate under all-or-nothing (0/1) loss. [4] [Total 6] 5 The annual number of claims from a particular risk has a Poisson distribution with mean /. The tyle prior distribution for # has a gamma distribution with a = 2 and 1 = 5. Claim numbers X1...., x,, over the last n years have been recorded. (i) Show that the posterior distribution is gamma and determine its parameters. [3) (ii) Given that n =8 and x; =5 determine the Bayesian estimate for / under: i-1 (a) squared-error loss (b) all-or-nothing loss (c ) absolute error loss. [5] [Total 8]