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5. Consider A Population With An Unknown Mean U And Finite Variance O2. (a) Show That X Is A Consist

5. Consider a population with an unknown mean u and finite variance o2. (a) Show that X is a consistent estimator of p. Now consider a new estimator X* of u, defined as follows: V S X in? if Xi > X1 if X1 = X(1) That is, that we would choose X to be our estimator if the first member of our sample is not the smallest value, and we would choose na to be our estimator if the first member of the sample is the smallest value. (b) Find E (X*|X1 > X(1)] and E (X*|X1 = X(1)] (c) Use your answers from part (b) and the Law of Total Expectation to find E (X*]. Is X* an unbiased estimator of u, an asymptotically unbiased estimator of u, or neither? (Note, it will be useful to know that P(X = X(1) = 1/n. ) (d) Using conditional probability, determine if X* is a consistent estimator of u.