# 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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