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Problem 3. Let Y1, Y2, and Y3 be independent, identically distributed random variables from a population...

Problem 3. Let Y1, Y2, and Y3 be independent, identically distributed random variables from a population with mean µ = 12 and variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 + Y3) denote the average of these three random variables.

A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an unbiased estimator of µ?

B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?

C. Consider a different estimator of µ: W = 1/8 Y1 + 3/8 Y2 + 1/2 Y3 This is an example of a weighted average of the Yi. What is the expected value of W, i.e., E(W) =? Is W an unbiased estimator of µ?

D. Find the variance of W, i.e., Var(W) =?

E. Consider another estimator of µ: V = 1Y1 + 0Y2 + 0Y3 This estimator only uses Y1 and completely ignores Y2 and Y3. What is the expected value of V, i.e., E(V ) =? Is V also an unbiased estimator of µ?

F. Find the variance of V, i.e., Var(V ) =?

G. Based on your answers in previous parts, which estimator of µ do you prefer? Please explain

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