Consider a population X which follows a uniform distribution U(0, θ) with θ > 0, and let X1, X2, ..., Xn be a random sample of X. Let M = max{X1, X2, ..., Xn}, the maximum value observed in the sample.
a) Use the fact that P(M ≤ t) = P(X1 ≤ t, X2 ≤ t, ..., Xn ≤ t) to determine the sampling distribution of M.
b) It is shown that M is the maximum likelihood estimator of θ. Determine the root mean square error of this estimator as a function of θ and n.