Question 2 The random variables X1; X2, ...; Xan are independent and normally distributed with common variance o'. However, X1; X2, ..., Xn have mean 0 while Xntl, Xn+2, ..., Xan have mean /. (a) Write down the joint pdf of X1, X2, ...; Xen and hence the likelihood function and log-likelihood of ( 1, 02 ). (b) Show that the maximum likelihood estimators of a and of are 2n 2n a = = ) x, and a? = 2n 7=n+1 Note: You do not need to check if the stationary point is a maximum. (c) Derive the Cramer-Rao lower bounds for the variances of unbiased estimators of 7 (#, a?). (d) Briefly explain whether a or o' have unbiased estimators that attain the relevant Cramer-Rao lower bounds. (e) For either maximum likelihood estimator in (6) above that is biased, find the unique minimum variance unbiased estimator, and explain why it is so