Please provide accurate answers for the following. Thank you

Annual numbers of claims on three different types of insurance policy follow a Poisson distribution with parameter , for i = 1, 2, 3. Data for the last four years is given in the table below. Year Type Total 11 AAS W N - 16 20 Derive the maximum likelihood estimate of p and calculate the corresponding estimates of uz and uj- [5] (ii) Test the hypothesis that 1, , and u, are equal using the scaled deviance. [5] [Total 10] The following time scries model is being used to model monthly data: where e, is a white noise process with variance of. (i) Ferform two differenceng transformations and show that the result is a moving average process which you may assume to be stationary. [3] (ii) Explain why this transformation is called seasonal differencing [1] (iii) Derive the auto-correlation function of the model generated in part (i). [8] ['Total 12] The number of claims, /, in a given year on a particular type of insurance policy is given by: P(N = n) = 0.8 x 0.2" n = 0, 1, 2. ... Individual claim amounts are independent from claim to claim and follow a Parcto distribution with parameters o - 5 and 2 - 1,000. (i) Calculate the mean and variance of the aggregate annual claims per policy. [4] (ii) Calculate the probability that aggregate annual claims exceed 400 using: (@) a Normal approximation. a Lognormal approximation. [G] (iii) Explain which approximation in part (ii) you believe is more reliable. [2] [Total 12]