2. Consider a large population which has true mean μ and true standard deviation . We take a sample of size 3 from this population, thinking of the sample as the RVs X1,X2,X3 where Xi can be considered iid. We are interested in estimating μ .
(a) Consider the estimator μ1 = X1 + X2 − X3 . Is this estimator unbiased? Calculate the bias.
(b) Find the variance of μ1 .
(c) When estimating μ , explain why someone would prefer the estimator ¯X = (X1+X2+X3)/3 over μ1 .
(d) Now consider the estimator μ2 = (X1+X2+X3)/2 . Is this estimator unbiased? Calculate the bias.
(e) Compute the MSE for μ2 .
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