Consider two independent random variables $X$ and $Y$ representing test scores from two different exam sections. Suppose $Var(X) = \sigma_X^2$ and $Var(Y) = \sigma_Y^2$. A student defines a composite performance index $Z = \alpha X + (1-\alpha) Y$, where $0 < \alpha < 1$. To minimize the variance of the composite index $Var(Z)$, what value of $\alpha$ should be chosen?