Please help me solve the following questions correctly.,,,

3. Assume so glad. draws from Y ~ Ezpmentiaiw) with pdf: 1 1 felt?) = germ-3y) for y > 0 E(Y|6) = a; var(Y|6)=62 and consider the use of s2 = E (K- - z/(n q 1) as an estimator of var(Y|6) = 62. 21 (a) It is well known that the sample variance is an unbiased estimator for the pop- ulation variance, and for the exponential distribution it is also the case that easz) = % [9 El] . Use these two pieces of information to explain why the sufcient conditions for s2 to be a consistent estimator of 92 hold. 2 (b) Use M = 5000 independent samples of size n = 100 to estimate E (32) and ends\") from an exponential distribution with 6 = 5. Report your estimates, compare than with their theoretical values, and also provide kernel estimates of the distribution of your simulated 32 values. Repeat this exercise with n = 1000, and then again with n = 10000. (c) Outline how the results found in part (b) provide numerical demonstration of the consistency of 52. A credibility factor represents the proportion of a posterior estimated parameter value that can be attributed to a (sensible) prior estimate of the same parameter. Select one: O True O False Under a frequentist approach, we have Ex (0 - 0(X))2 = Bias' + Variance The "Variance" here refers to the population variance, given by Var(Xi). Select one: O True O FalseQuestion 3. [3 point3]Consider the mode] K- : n+3, 3' = 1, . . . , n, Where KS are independent. Find the LS estimate of pi. Also, nd the variance of your estimate. Here, BE) 2 0 and V3r(g) : 02 An insurance company is reviewing claim amounts arising from a particular portfolio of insurance policies over the past 3 years. The data is split into four different regions of England. Year Region 2012 2013 2014 South 1500 2000 2050 East 1200 1500 1 800 Midlands 1000 1100 1200 North 1 100 1600 1800 a. Using Empirical Bayes Credibility Theory model 1 (EBCT model 1) estimate the pure premium for all four regions in 2015. b. The total number of policies sold for this particular portfolio of insurance policies are Year Region 2012 2013 2014 South 15 20 East Midlands 12 North 11 18 In 2015 25 policies are sold in the South region. Analyse the data using Empirical Bayes Credibility Model 2 (EBCT Model 2), and calculate the pure premium in 2015 for the South region