Let X1, X2, , Xn be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is a = max (Xi).
(a) Argue intuitively why cannot be an unbiased estimator for a.
(b) Suppose that E(a) = na/(n + 1). Is it reasonable that consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large.
(c) Propose an unbiased estimator for a.
(d) Let Y = max (Xi). Use the fact that if and only if each to derive the cumulative distribution function of Y. Then show that the probability density function of Y is

Use this result to show that the maximum likelihood estimator for a is biased.

ny"-1 f(y) = 0sysa otherwise