Question: Suppose that X1 and X2 are independent random variables having
Suppose that X1 and X2 are independent random variables having a common mean μ. Suppose also that Var(X1) = σ21 and Var(X2) = σ22. The value of μ is unknown, and it is proposed that μ be estimated by a weighted average of X1 and X2. That is, λX1 + (1 − λ)X2 will be used as an estimate of μ for some appropriate value of λ. Which value of λ yields the estimate having the lowest possible variance? Explain why it is desirable to use this value of λ.
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