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Let x1, x2, ... , xn be a random sample from a population with mean mu and variace sigma-squared. Let X bar be the sample mean and S-squared be the sample variance. Prove that Xbar squared is a biased estimator for mu-squared. Find the value of k so that the estimator xbar-squared minus kS-Squared is an unbiaased estimator of mu-squared.


|||1) For Xbar we know that


* E(Xbar) = 碌


* Var(Xbar) = sigma虏/n


Since Var(A) = E(A虏) - E虏(A), we have E(A虏) = Var(A)+E虏(A)


Hence


* E((Xbar)虏) = Var(Xbar)+E虏(Xbar) = sigma虏/n + 碌虏


Since this is not equal to 碌虏, (Xbar)虏 is biased





2) For s虏 we know that E(s虏) = sigma虏


We have


E((Xbar)虏 - ks虏) = sigma虏/n + 碌虏 - k*sigma虏


If we choose k = 1/n, we find ... = 碌虏