Answer all the questions

2. Let X - {X1, X2. . . ., X} be a random sample drawn from the distribution with probability density function: fx (1: 0) = otherwise. (a) Derive the probability density function of X(m) - max(X1, X2, . . ., X,.). (5 marks) (b) Find a sufficient statistic for 0. (3 marks) (c) Derive the maximum likelihood estimator of 0 and verify that it is a biased estimator of 0. (5 marks) (d) Propose an unbiased estimator of 0 and check whether this proposed estimator is mean-square consistent. (7 marks)