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 = 71 Xi We know that the sample mean

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.