Answer all the questions

function: 3. Let X = {X1, X2, ..., X,} be a random sample from the probability density fx (:I; B) = - 681 T' exp (-3 for r > 0 and 0 otherwise, where B > 0 is an unknown parameter. You are given that: E(X) - 43 and Var (X) = 482. (a) Derive the likelihood Linetion, the score function and the Fisher information for B. (6 marks) (b) What is the Cramer Rao lower bound for the variance of an unbiased estimator of B? (2 marks) (c) Find the maximum likelihood estimator of B. Is this the minimum variance unbiased estimator of B? Justify your answer. (6 marks) d) You are told that the sample size of the random sample is n - 15 and the sample mean is z - 7.5. Provide a 95% confidence interval for B. If we increase the value of r, and keep everything else constant, what will happen to the expected length of your provided interval? Justify your answer. (Hint: If Z ~ N(0, 1), then P(Z - 1.96) ~ 0.025.) (6 marks)