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Given $X$ and $Y$ be jointly normally distributed with $\mu_X=20, \mu_Y=40,\sigma_X=3, \sigma_Y=2$ and $\rho=0.6.$ Find the mean and the variance of $U=X+Y.$ soln: $U\sim N(\mu=60,\sigma^2=13).$ Am I right?

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  • $\begingroup$ You haven't taken into account $\rho=0.6$ at all. $\endgroup$ – Michael Hardy Mar 22 '18 at 21:52
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Hint: Your result is not right. The variance of the sum of two normally distributed random variables is:

$$\sigma_{X+Y}^2=\sigma^2_U=\sigma_X^2+\sigma_Y^2+2 \rho\sigma_X \sigma_Y$$

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