an unbiased estimator for the population mean μ? Explain.b. Would it still be unbiased if the coin was not fair?c. Compute the variance of the x-bar estimator.d. Which one is a better estimator of μ, or the “gambler’s one? Explain why.2. Suppose we have a i.i.d sample X1, X2, …, Xn, with E(Xi) = u, and Var(X;) = -, i = 1,2, …, n. Recall_X=71XiWe know that the sample mean X is an unbiased estimator of u.Now suppose that I flip a fair coin (i.e. P(H) = P(T) = 1/2), and I propose a new estimatorfor the population mean u. This new "gambler’s estimator" for the mean is given by:X = X+Dwhere D = +1 if the flip is heads and D = -1 if the flip is tails.a. Is X an unbiased estimator for the population mean u? Explain.b. Would it still be unbiased if the coin was not fair?c. Compute the variance of the X estimator.d. Which one is a better estimator of u, X or the "gambler’s one? Explain why.




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