# Minimizing the variance

Let X,Y be random variables such that X has variance 2, Y has variance 3 and the correlation coefficient between X and Y is equal to 0.45.

Find a real number a which minimizes the variance of X + a⋅Y

I've already found the variance for the first part (Var(X + Y) = 7.2045407685), and I think that the Var(x + a⋅Y) = Var(X) + a^2 * Var(Y) + 2 * Cov(X,Y) but could someone please explain to me what "minimizing the variance" means and where to begin? Thanks!

• just compute the variance of X + aY in terms of a; differentiate the resulting function with respect to a, and minimize!!! – Frank Mar 20 '14 at 23:14
• I spaced on the differentiating part of it... thank you!! – Shell Mar 20 '14 at 23:26

I think you meant: $$Var(x + a⋅Y) = Var(X) + a^2 * Var(Y) + 2a * Cov(X,Y)$$ As Frank already said, substitute the values you know, differentiate with respect to $a$, set equal to $0$, and solve for $a$.