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[All Steps] Let Y 1 Y 2 Y 3 and Y 4 be independent, identically distributed random variables from a population with mean µ and variance σ ^2. Let

Question: Let Y 1 Y 2 Y 3 and Y 4 be independent, identically distributed random variables from a population with mean µ and variance \({{\sigma }^{2}}\). Let \(\bar{Y}\) = ¼( Y 1 + Y 2 +Y 3 + Y 4 ) denote the average of these four random variables.

  1. What are the expected value and variance of \(\bar{Y}\) in terms of mean and variance?
  2. Now, consider a different estimator of µ: W = 1/8 Y 1 + 1/8 Y 2 +1/4 Y 3 +1/2 Y 4 This is an example of a weighted average of the Y i . Show that W is also an unbiased estimator of µ. Find the variance of W.
  3. Based on your answers to parts a and b, which estimator of µ do you prefer, Ybar or W?

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