Question: Suppose X₁ and X₂ are independent random variables with the same unknown mean µ_X. Both X₁ and X₂ have variance of 25. Let T=aX₁ + bX₂ be an estimator of µ_X, where a and b are constants.

a) Show that, if \(a+b=1\), T is an unbiased estimator of µ_X.

b) If a=¼ and b=¾, what is the variance of T?

c) If a=½ and b=½, what is the variance of T?

d) 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.)