Minimizing Risk

Probability Level 3

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?