Let X1, X2, .., An be a random sample from a distribution with probability density function In x 1 if 0 0; f (x0') = otherwise; Given: E (X) = : 2 ; and Var (1=]] -8. If f (x]o=) belongs to the regular 1-parameter exponential family of density functions, then it can be expressed as: (x) exp (er (x) - wej)- Futhermore, E (!(X)) = w (@) and Var (:(X)) = w"(@). (a) Prove or disprove that f (xo) belongs to the regular 1-parameter exponential family of density functions. (6) (b) Use the factorization theorem/criterion to show that > In' X is a sufficient statistic for o'. (5) (c) Show that > In' X, is a minimal sufficient statistic for o'. (5) (d) Prove or disprove that the respective mean and variance of > In' X, are no ? and 2not. (7) (e) Use parts (a) and (d) to find a minimum variance unbiased estimator of o'. Determine the variance of the estimator. (8) (f) Show that the maximum likelihood estimate of ' is the same as the minimum variance unbiased estimate of - in part (e). (5) (g) What are the maximum likelihood estimates of o and E (X)? Justify your answers. (5)(a) Derive the method of moments estimator of &?. (3) (b) Consider the following Taylor's series expansion 2mXxo'+le 2 X-62 about E (X) =e 2 where I = - > Xi. Prove or disprove that 2 In I is a consistent estimator of oz. (7) (c) Find: (i) the Fisher information for o' in the sample; and (5) (ii) the Cramer-Rao lower bound of the variance of an unbiased estimator of c. (4) (d) Use another method (beside the one you used in QUESTION 1 (e)) to prove that the maximum likelihood estimator of o' that you found in QUESTION 1 (f) is also a minimum variance unbiased estimator of oz (5) (e) Write down an expression for the approximate 95% confidence interval for - in terms of the maximum likelihood estimate of o