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Let * be a random variable with density /(x) = ed for x > 0. (i) Construct an algorithm for generating random samples from X [2] A sequence of simulated observations is required from the density function h(x ) =2xe-* x30. (ii) Construct a procedure using the Acceptance-Rejection method to obtain the required observations. [5] (iii) Calculate the expected number of pseudo-random numbers required to generate 10 observations from h using the algorithm in part (ii). [2] [Total 9] An insurance company has for five years insured three different types of risk. The number of policies in they'll year for the " type of risk is denoted by P; for i = 1. 2. 3 and / = 1, 2, 3, 4, 5. The average claim size per policy over all five years for the th type of risk is denoted by X; The values of P, and X, are tabulated below. Number of policies Mean claim size Risk type i Year 1 Year 2 Year 3 Year 4 Year 5 X; 17 23 21 29 35 850 42 51 60 35 37 720 43 31 62 98 107 900 The insurance company will be insuring 30 policies of type 1 next year and has calculated the aggregate expected claims to be 25,200 using the assumptions of Empirical Bayes Credibility Theory Model 2. Calculate the expected annual claims next year for risks 2 and 3 assuming the number of policies will be 40 and 110 respectively. [9]