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Claims from a portfolio are believed to have a Parero(c, 1) distribution. In Year , 4=6 and 1 =1,000. An excess of loss reimurance arrangement is in force, with a retention limit of 500. Inflation is a constant 10% pa. Find the distribution of the insurer's claim payments in Years 1 and 2 before reinsurance. () Find the percentage increase in i net claims payout in each year.Claims in a portfolio are believed to arise as an Exp(1) distribution. There is a retention limit of 1.000 in force, and claims in excess of 1.000 are paid by the reinsurer. The insurer, wishing to estimate A , observes a random sample of 100 claims, and finds that the average amount of the 90 claims that do not exceed 1,000 is 82.9. There are 10 claims that do exceed the retention limit. Find the MLE for 1.A statistician wishes to find a Bayesian estimate of the mean of an exponential distribution with density function f(x)=-e""#. He is proposing to use a prior distribution of the form: prior (4) - "+T(a) You are given that the mean of this distribution is (-1 (i) Write down the likelihood function for / , based on a random sample of values 4,A, from an exponential distribution. [1] (ii) Find the form of the posterior distribution for /, and hence show that an expression for the Bayesian estimate for / under squared error loss is: D = [4] n+a-1 (iii) Show that the Bayesian estimate for / can be written in the form of a credibility estimate. and write down a formula for the credibility factor. (3] (iv) The statistician now decides that he will use a prior distribution of this form with parameters 0-40 and or-1.5. His sample data have statistics a -100, Ex; = 9,826, and Ex, =1,200,000. Find the posterior estimate for #, and the value of the credibility factor in this case. [3] Comment on the results obtained in part (iv). [Total 13]