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[See Solution] Suppose X 1 and X 2 are independent random variables with the same unknown mean µ X . Both X 1 and X 2 have variance of 25. Let T= aX 1 + bX

Question: Suppose X 1 and X 2 are independent random variables with the same unknown mean µ X . Both X 1 and X 2 have variance of 25. Let T= aX 1 + bX 2 be an estimator of µ X , where a and b are constants.

  1. Show that, if \(a+b=1\), T is an unbiased estimator of µ X .
  2. If a=¼ and b=¾, what is the variance of T?
  3. If a=½ and b=½, what is the variance of T?
  4. What choice of a and b minimizes the variance of T, subject to the requirement that T be an unbiased estimator of µ X ? (Hint: Find the variance of T and use the fact that T is unbiased to write the variance in terms of only a. Then find the value of a that minimizes the variance.)

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Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


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