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5. Let X1, X2, …, Xn be n uncorrelated observations of the random variable X, which has a distribution with population mean E(X) and variance 2. For any linear estimator W of E(X), we can write W = a1X1 + a2X2+ … + anXn, where the ai are all constants (numbers). A) What restriction on the ai terms is needed for W to be an unbiased estimator of E(X)? (1 point) a1 needs to be a constant and sum up to 1 b) At a party this Friday night, one of your friends proposes the following estimator for E(X) for a sample of 4 people: W = (1/2)*X1 +(1/6)*X2+ (1/6)*X3 + (1/6)*X4 Is this estimator unbiased? What is the variance of this estimator, and is it efficient? (Hint: what is the variance of the sample mean?) What do you tell your friend about this idea?
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