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An actuary has heen advised to use the following positively-skewed claim size distribution as a model for a particular type of claim, with claim sizes measured in units of El(0. f(x;0) = exp UU with moments given by EIX] = 30. EX-1 = 120- and E[X3] = 6063. (1) Determine the variance of this distribution and calculate the coefficient of skewness. [4] (ii) Let X1. X2. .... X, be a random sample of a claim sizes for such claims. Show that the maximum likelihood estimator (MLE) of 0 is given by 0 = and show that it is unbiased for C. [5] (Hii) A sample of n = 50 claim sizes yields Ex, =313.6 and Ex- = 2,675.68. (a) Calculate the MLE 0. (b) Calculate the sample variance and comment briefly on its comparison with the variance of the distribution evaluated at 0. (c) Given that the sample coefficient of skewness is 1.149, comment briefly on its comparison with the coefficient of skewness of the distribution. [4] (iv) (a) Write down a large-sample approximate 95%% confidence interval for the mean of the distribution in terms of the sample mean $ and the sample variance s. IIence obtain an approximate 95% confidence interval for 0 and evaluate this for the data in part (ii) above. (b) Evaluate the variance of the distribution at both the lower and upper limits of this confidence interval and comment briefly with reference to your answer in part (ifi)(b) above. [5] [Total 18](i) Explain why a mortality experience would need to be graduated. [3] An actuary has conducted investigations into the mortality of the following classes of lives: (a) the female members of a medium-sized pension scheme (b) the male population of a large industrial country (c) the population of a particular species of reptile in the zoological collections of the southern hemisphere The actuary wishes to graduate the crude rates. (ii) State an appropriate method of graduation for each of the three classes of lives and, for each class, briefly explain your choice. [Total 6] A study of the mortality of a certain species of insect reveals that for the first 30 days of life, the insects are subject to a constant force of mortality of 0.05. After 30 days, the force of mortality increases according to the formula: H30+x =0.05 exp(0.01x) . where x is the number of days after day 30. (i) Calculate the probability that a newly born insect will survive for at least 10 days. [1] (ii) Calculate the probability that an insect aged 10 days will survive for at least a further 30 days. [3] (Hii) Calculate the age in days by which 90 per cent of insects are expected to have died. (4] [Total 8] (i) Define a counting process. [2] For each of the following processes: simple random walk compound Poisson Markov chain (li) (a) State whether each of the state space and the time set is discrete, continuous or can be either. (b) Give an example of an application which may be useful to a shopkeeper selling dried fruit and nuts loose. [6] [Total 8]