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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!

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

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  • $\begingroup$ Thank you! I appreciate it $\endgroup$ – Shell Mar 20 '14 at 23:19

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