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Suppose we have two random variables that follow a multivariate normal distribution,

$[x,y]\sim MVN([a,b], \begin{bmatrix} \rho_1 & \rho_3 \\ \rho_3 & \rho_2 \end{bmatrix})$

Then what is the distribution for the sum of x and y?

That is,

$z=x+y \sim N(?,?)$

Surely, $N(a+b,\rho_1+\rho_2)$ does not seem right, since we need to consider the correlation between x and y. But what is the correct way to incorporate this correlation aspect into the final distribution?

Ding Li
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    Typing in "variance of the sum of two random variables" in your favorite search engine should quickly give you the answer. – statmerkur Jun 14 '22 at 09:56
  • https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables#Correlated_random_variables has the answer. – whoknowsnot Jun 14 '22 at 10:12

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