Start this question on a new A4 sheet. Show all necessary calculation steps. Where applicable, you should use ONLY the definitions given in your lectures, and not definitions from any external sources. All numerical answers should be rounded to TWO decimal places. For example, 0.6783 is rounded to 0.68, and 0.298 is rounded to 0.30. No rounding is needed for numerical answers of the forms, say 3 and 2.8. Question 4 (10 marks) Though we mainly focused on Method of Moments estimators (MoM) and Maximum Likelihood Estimators (MLE) in the context of continuous random variables, the same ideas and calculations apply for discrete random variables. This will be explored in the following question. Let X, be the number of claims in a particular line of insurance business in Year i, for i = 1, ...,5. Assume that X1, ..., As are independent and identically distributed (iid) Poisson random variables with unknown parameter > > 0, where A is the expected number of claims in the insurance business line per year. For your convenience, the probability function (or the probability mass function) of the Poisson distribution with unknown parameter ) is given by: fx, (r; ) ) = = I! Now, suppose that the numbers of claims in the insurance business line for these five years are given as follows: I = (T1, 12, T3, TA, 15) = (2, 1, 0, 6, 3). Using the above given data on the numbers of claims, answer the questions in Part a) and Part b) as follows: a) Use the first moment to find the Method of Moments estimator for A. (2 marks) b) Determine the likelihood function L(); z) as a function of 1. (3 marks) c) Determine the log-likelihood function log L(); r) as a function of 1, (note that log is the natural logarithm). (2 marks) It can be shown that the MLE of A is A(MLE) = = ) Xi, say the sample mean of the X's, (you do not need to prove this). d) Calculate the bias of A(MLE) as an estimator of 1. (2 marks) e) Is A(MLE) an unbiased estimator of A? Justify your answer. (1 mark)