Let (X1,..., Xn) be a random sample of binary random variables with P(X1 = 1) = p E (0, 1). (i) Obtain the Bayes estimator of p(1 p) when the prior is the beta dis- tribution with known parameter (a, B), under the squared error loss. (ii) Compare the Bayes estimator in (i) with the UMVUE of p(1 p). (iii) Discuss the bias, consistency, and admissibility of the Bayes estimator in (i). (iv) Let [p(1 p)]-I(0,1) (P) be an improper Lebesgue prior density for Show that the posterior of p given X;'s is a probability density provided that the sample mean X E (0, 1). (v) Under the squared error loss, find the generalized Bayes estimator of p(1 p) under the improper prior in (iv). .