please provide a precise solution

Give two examples of exercises where Monte-Carlo simulation should be performed using the same choice of random numbers, explaining your reasoning in each case. (4] An insurance company has collected data for the number of claims arising from certain risks over the last 10 years. The number of claims in the jth year from the ith risk is denoted by Xy for i = 1, 2, ..., # and / = 1, 2, ..., 10. The distribution of X; for = 12, ..., 10 depends on an unknown parameter 0; and given 0; the X, are independent identically distributed random variables. (1) Give a brief interpretation of E[s'(0)] and VIm(0))] under the assumptions of Empirical Bayes Credibility Theory Model I. 121 (ii) Explain how the value of the credibility factor 7. depends on F[s-(4)] and Vm(0)]. [3] [Total 5] Let y1, -... >'n be samples from a uniform distribution on the interval [0, 0] where 0 3 0 is an unknown constant. Prior beliefs about A are given by a distribution with density 0 otherwise where a and f are positive constants. (i) Show that the posterior distribution of U given y is of the same form as the prior distribution, specifying the parameters involved. [4] (ii) Write down the posterior distribution of 0 given y . ....),. 121 [Total 6] The annual number of claims on an insurance policy within a certain portfolio follows a Poisson distribution with mean u. The parameter u varies from policy to policy and can be considered as a random variable that follows an exponential distribution with mean Find the unconditional distribution of the annual number of claims on a randomly chosen policy from the portfolio. [6]