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1 A motor insurance portfolio produces claim incidence data for 100,000 policies over one year. The table below shows the observed number of policyholders making 0, 1, 2, 3, 4, 5, and 6 or more claims in a year. No. of claims No. of policies 87,889 11,000 2 1,000 100 10 Un 1 26 Total 100,000 (i) Using the method of moments, estimate the parameter of a Poisson distribution to fit the above data and hence calculate the expected number of policies giving rise to the different numbers of claims assuming a Poisson model. [3] (ii) Show that the estimate of the Poisson parameter calculated from the above data using the method of moments is also the maximum likelihood estimate of this parameter. [4] (iii) Using the method of moments, estimate the two parameters of a Type 2 negative binomial distribution to fit the above data and hence calculate the expected number of policies giving rise to the different numbers of claims assuming a negative binomial model. [6] You may use the relationship: P(X = x) = k+x-1 xqxP(X =x-1) X for the negative binomial distribution. (iv) Explain briefly why you would expect a negative binomial distribution to fit the above data better than a Poisson distribution. [2] [Total 15]A random sample X1..... X,, is taken from a normal distribution with mean / and variance of (1) State the distribution of 2 (X, - x)2 [1] It is decided to estimate the variance, o, using the following estimator: 62 = 1 n+b ZIX,- XP where b is a constant. (ii) (a) Use part (i) to obtain the bias of a . (b) Hence, show that & is unbiased when b=-1. [3] (iii) (a) Show, using parts (i) and (ii)(a), that the mean square error of is given by: MSE(G2)-2(0-1)+ (1+6) (n + b) ? (b) Determine whether the estimator, @