statistical questions

Consider an i.i.d. sample X1, ..., X, from the continuous probability distribution D(#) with density function f(x) = ce(x-M)> p- 1Exsp+1, where # E (-co, co) is an unknown parameter. Throughout this question, c denotes the fixed constant c= edx) =0.341845. (a) In the special case where # = 0, explain why EX, = 0. Hence or otherwise determine the expected value of the distribution D(p) for general # E (-00, co). 3 pts (b) Find the method-of-moments estimator AMM of #. Explain why is it unbiased. 3 pts (c) Show that Var(X]) = ce - }- 4 pts (d) What is the mean-squared error of AMM as an estimator for ? 2 pts (e) Find an approximate 95% confidence interval for # given that we observe a sample of size n = 50 with sample mean x = 3.43 and sample variance s = 0.40. 4 pts (f) Show that the likelihood function of the parameter , with observations x1, . .., In is L( p; X1, . . . . XM) =Ce(-4)2 max x - 1