Question: Suppose you have a sample X1, X2, ...,.Xn (with n ≥ 3) from a population of unknown distribution, with mean \(\mu \) and standard deviation \(\sigma \) (and therefore variance \({{\sigma }^{2}}\) ). While you do not know the population distribution, you do know that each X; is independent and identically distributed (i.i.d.).

1. What is E[X2]?
2. Suppose I create an estimator \(\bar{X}\) for the population mean ( \(\mu \) ), which I define as: \(\bar{X}=\frac{{{X}_{1}}+{{X}_{2}}}{2}\) . That is to say, my estimator calculates the sample mean using only the first two observations. Showing your work, calculate E[X].
3. Is \(\bar{X}\) an unbiased estimator for the population mean \(\mu \) ? Explain.
4. Showing your work, calculate Var[X].
5. Now consider the traditional estimator, \(\bar{X}=\frac{1}{n}\sum{{{X}_{i}}}\). Showing your work, calculate Var[X].
6. Compare the variance of X to that of X, recalling that we know that n ≥ 3. Which one is larger, and why?
7. (Extra Credit) Is \(\bar{X}\) a consistent estimator for the population mean \(\mu \) ?

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