### Description

1. Let X1, …, Xn be i.i.d. with the pdf f(x; ?) = ? x??1 , 0 ? x ? 1, ? > 0. (a) Write down the likelihood function and find the MLE of ?. (b) Find the MLE of ? = P?(X ? 0.5). (c) Find the moment estimator of ?. 2. Let X1, …, Xn be i.i.d. with the pdf f(x; ?) = ? x ?+1 , x ? 1, and f(x; ?) = 0 elsewhere. (a) Write down the likelihood function and find the MLE of ?. (b) Find the MLE of ? (?) defined by P?(X ? ? (?)) = 0.95. (c) Find the moment estimator of ?. 3. Let X1, …, Xn be i.i.d. with the gamma distribution f(x; ?) = xe?x/? ? 2 , x > 0, where ? > 0. (a) Find the MLE of ?. (b) Calculate the bias, variance and MSE of the MLE. (c) Is the MLE MSE consistent? Answer yes or no and explain. (d) Find the Fisher information I(?). (e) Is the MLE a UMVUE? Answer and explain. 4. Consider a random sample X1, …, Xn from the geometric pdf f(x|?) = ? (1 ? ?) x?1 , x = 1, 2, 3, … where 0 < ? < 1. Assume that ? has the uniform prior p(?) = 1 , 0 ? ? ? 1 and p(?) = 0 otherwise. (a) Find the posterior distribution of ? given the sample. (b) Find the Bayes estimator of ? (under squared error loss).