Please solve the question in the attachment below with precision

Claims on a portfolio of insurance policies arise as a Poisson process with parameter 2. Individual claim amounts are taken from a distribution Y" and we define m; = E(X') for / = 1, 2, .... The insurance company calculates premiums using a premium loading of 0. (1) Define the adjustment coefficient R. [1] (ii) Show that R can be approximated as 2041 , by truncating the series expansion of My(!). 1712 [3] Now suppose that ) follows an exponential distribution with parameter y. (iii) Show that R = (1+ 0) [3] The insurance company uses a premium loading of 12%. and the mean claim amount is 200. (iv) Calculate R, commenting on the difference with the approximation to R shown in part (ii). [3] The initial surplus is 5,000. (v) Calculate an upper bound for the ultimate probability of ruin. [1] (vi) Suggest two methods by which the insurance company can reduce the probability ofruin. [2] [Total 13] For a portfolio of insurance policies, claims X, are independent and follow a gamma distribution, with parameters u = 6 and p, which is unknown. A random sample of a claims, X,..... X,, is selected, with mean X . (i) Derive an expression for the estimator of B using the method of moments. [2] (ii) Explain what the Maximum Likelihood Estimator (MLE) of B represents. [2] (iii) Derive an expression for the MLE of p, commenting on the result. [5] (iv) State the Moment Generating Function (MGF) of X. L1] Let Y = 2nRY (v) Derive the MGF of Y, and hence its distribution, including statement of parameters. LS] [Total 15]