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A portfolio of insurance policies contains two types of risk. Type I risks make up 80%% of claims and give rise to loss amounts which follow a normal distribution with mean 100 and variance 400. Type II risks give rise to loss amounts which are normally distributed with mean 115 and variance 900. (i) Calculate the mean and variance of the loss amount for a randomly chosen claim. [3] (ii) Explain whether the loss amount for a randomly chosen claim follows a normal distribution. [2] The insurance company has in place an excess of loss reinsurance arrangement with retention 130. (iii) Calculate the probability that a randomly chosen claim from the portfolio results in a payment by the reinsurer. [3] (iv) Calculate the proportion of claims involving the reinsurer that arise from Type II risks. [2] [Total 10] Consider the time series Y, =0.1 +0.41+ 0.9erite, where e, is a white noise process with variance of. (i) Identify the model as an ARIMA(p,d,q) process. [1] (Hi) Determine whether Y, is: (7) a stationary process (b) an invertible process [2] (iii) Calculate E(Y ) and find the auto-covariance function for Y,. (6] (iv) Determine the MA(30) representation for Y. (4] [Total 13] Claim events on a portfolio of insurance policies follow a Poisson process with parameter 2. Individual claim amounts follow a distribution X with density Ax) = 0.012xe-001x The insurance company calculates premiums using a premium loading of 45. (i) Derive the moment generating function My(). [3]