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(7) 112 ptsl Let Xi,..., XT denote a random sample of size T from X, where VIX] < oo. a +bX, and for each t define Zt a +bX, for some Define a new random variable Z constants a and b. (a) Show that Z = a + bX and 03-b2q, where the sample me an X and sample variance x of the original sample are as defined in class (b) Prove that Z is an unbiased estimator of E[Z] (c) Prove that Z is a consistent estimator of ElZ]. Be sure to state clearly what theorems you are invoking in your answer. (HINT: use the fact that g(t)-a +bt is a continuous inction.)

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