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An actuary is simulating claim sizes, X, for a particular insurance policy. X follows an exponential distribution with varying parameter ), where A can take one of three possible values. The table shows the distribution function of this external factor, and its impact un h. Probability 0.3 0.3 0.4 1 2 3 Set out an algorithm to generate samples from X. using the inverse transform method. [3] (i) State the three main components of a generalised linear model. [3] Consider the discrete random variable Y, with the following probability density function =....,1 (ii) Show that Y' belongs to the exponential family of distributions, specifying each component. [4] (iii) State the canonical link function in this case. [1] [Total 8] (i) State the fundamental difference between Bayesian estimation and Classical estimation. (ii) State three different loss functions which may be used under Bayesian estimation, indicating for each its link to the posterior distribution. [3] The proportion, O, of the population of a particular country who use online banking is being estimated. Of a sample of 500 people, 326 do usc online banking. An actuary is estimating O using a suitable uniform distribution as a prior. (iii) (a) Determine the posterior distribution of e. (b) Calculate an estimate of o using the loss function that minimises the mean of the posterior distribution. [4] [Total 9]