Address the following questions effectively. All the parts please

A model used for claim amounts (X. in units of (10,000) in certain circumstances has the following probability density function, fix), and cumulative distribution function, F(r): fax= (10) 10 5. *30; F(x)=1- (10+ x)" 10+x You are given the information that the distribution of X has mean 2.5 units ((25,000) and standard deviation 3.23 units (632,300). (1) Describe briefly the nature of a model for claim sizes for which the standard deviation can be greater than the mean. [2] (Hi) (a) Show that we can obtain a simulated observation of X by calculating * = 10 (1-7)-02-1] where r is an observation of a random variable which is uniformly distributed on (0.1). (b) Explain why we can just as well use the formula *= 10 -02 -1] to obtain a simulated observation of X. (c) Calculate the missing values for the simulated claim amounts in the table below (which has been obtained using the method in (ii)(b) ahove): Claim (f) 0.7423 6,141 0.0291 10,2872 0.2770 29.272 0.5895 11.148 0.1131 54.635 0.9897 207 0.6875 7,782 0.8525 3.243 0.001G 0.5154 [5] [Total 7]An insurance portfolio contains policies for three categories of policyholder: A, B and C. The number of claims in a year, N, on an individual policy follows a Poisson distribution with mean A. Individual claim sizes are assumed to be exponentially distributed with mean 4 and are independent from claim to claim. The distribution of 2. depending on the category of the policyholder, is Category Value of 1 Proportion of policyholders 20% 60% 20% Denote by S the total amount claimed by a policyholder in one year. (i) Prove that E(S) = E[E(S A)]- [2] (ii) Show that E(S )) - 42. and Var(S |2) -322 . [2] (iii) Calculate E(S). (iv) Calculate Var(S). [Total 8] The following information is available for a motor insurance portfolio: The number of claims settled: Development Year Accident year 2006 442 151 50 2007 623 111 2008 581 The cost of settled claims during each year (in 000's): Development Year Accident year 0 2 2006 6321 1901 701 2007 7012 2237 2008 7278 Claims are fully run off after year 2. Calculate the outstanding claims reserve using the average cost per claim method with grossing up factors. Inflation can be ignored. [10]