Exercise 1

1. (25 points) Let X1, . .., Xn be a random sample of size n from the Bernoulli distribution P(X = x) = p"(1 -p)l-", x = 0,1. . (a) (7pts) What is the best unbiased (UMVUE) estimator for p and justify your answer? . (b) (6pts) Let W(X1, X2, X3) = X1X2X3, find a best unbiased estimator of T(p) =p'. . (c) (6pts) Find a uniformly most powerful (UMP) level a test of Ho : p = 1/3 versus H1: p = 2/3. . (d) (6pts) Find a 95% confidence interval estimator for the p