The distribution of claim size, X (in thousands of pounds), for a certain portfolio is modelled by the following probability density function f(x) = -x/2 = +xex2) x20. (i) Find the moment generating function for the claim size X. (ii) Obtain the mean, α2, variance and α3 for the claim size X. (iii) (iv) (v) The risk of claims for this portfolio is said to be homogenous with an average number of claims per year of 75. Find the mean, variance and skewness coefficient of the total annual claim size. The company applies a 6% safety loading for the policies in the portfolio. Estimate the initial reserve for the portfolio that would ensure that the total claims S for one year are covered with a probability of 99.95%, using skewness correction. If the company starts with an initial reserve of £125,000 find out the probability of meeting all claims in one year, using a skewness correction.

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