Standard deviation of a correlated sum

EasyRandom Variables Statistics~3m

Random variables $X$ and $Y$ have variances $9$ and $16$ respectively, with correlation $\rho(X,Y) = -\tfrac{3}{8}$ . Find the standard deviation of $X + Y$ .

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$\operatorname{Var}(X+Y)=\sigma_X^2+\sigma_Y^2+2\rho\,\sigma_X\sigma_Y$ with $\sigma_X=3,\ \sigma_Y=4$ .
$\operatorname{Cov}=-\tfrac38\cdot12=-4.5$ , so $\operatorname{Var}=25-9=16$ and the SD is $4$ .

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Asked at: Citadel, Two Sigma

Standard deviation of a correlated sum

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