Suppose that X1 and X2 are independent random variables having a common mean Suppose also that Var(X1) 21 and Var(X2) 22 The value of is unknown, and it is proposed that be estimated by a weighted average of X1 and X2 That is, X1 (1 )X2 will be used as an estimate of for some appropriate value of Which value of yields the estimate having the lowest possible variance Explain why it is desirable to use this value of

Question: Suppose that X1 and X2 are independent random variables having a common mean . Suppose also that Var(X1) = 21 and Var(X2) = 22. The

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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