Minimizing Risk

The prices (in dollars) of two stocks are random variables \(X\) and \(Y\) with \(E(X) = E(Y) = P.\) Estimators of these prices have variances of 4 and 8 respectively. Let \(\hat{P}\) be an unbiased estimator of \(P\) with

\[\hat{P} = aX + (1 - a)Y.\]

For what value of \(a\) is the variance of \(\hat{P}\) minimized?

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