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The number of claims, \\, which arise in a year on each policy of a particular class is to be modelled as a Poisson random variable with mean A. Let X = (X. X. .... X,) be a random sample of size a from the distribution of X, and let X =-> X;. I Suppose that it is required to estimate 1, the mean number of claims on a policy. (i) Show that 2, the maximum likelihood estimator of 2, is given by A= X . [3] (ii) Derive the Cramer-Rao lower bound (CRIb) for the variance of unbiased estimators of A. [4] (iii) (a) Show that 2. is unbiased for 1. and that it attains the CRIb. (b) Explain clearly why, in the case that n is large, the distribution of A can he approximated by (3] (iv) (a) Show that, in the case a = 100, an approximate 95% confidence interval for > is given by 7+ 0. 196 /Y (b) Evaluate the confidence interval in (iv)(a) based on a sample with the following composition: observation 2 3 frequency 11 28 19 28 [6] [Total 16](i) List four factors other than age and smoker status by which life insurance mortality statistics are often subdivided. [21 Two offices in different towns of the same life insurance company write 25-year term assurance policies. Below are data from these two offices relating to policyholders of the same age. Both deaths and policies in force are on an age last birthday basis. Gasperton Great Hawking Policies in force on 1 January 2009 2,000 1,770 Policies in force on 1 January 2010 2,100 1,674 Deaths in calendar year 2009 25 21 (ii) Calculate the central death rate for the calendar year 2009 at this age for the offices in Gasperton and Great Hawking. A detailed examination of the records shows that 50% of the policyholders in Gasperton at both censuses were smokers, and 20% of policyholders in Great Hawking at both censuses were smokers. National death rates at this age for smokers in 2009 were 40% higher than those for non-smokers. (iii) Estimate the central death rates for smokers and non-smokers in Gasperton and Great Hawking [4] The life insurance company charges policyholders in Gasperton and Great Hawking the same premiums for the 25-year terin assurance policies. It charges smokers in both towns 40% more than non-smokers. (iv) Comment on the company's pricing structure in the light of your results from parts (ii) and (iii) above. 131 [Total 11]