Minimizing Risk
Discrete Mathematics
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?
by
Andrew Ellinor