$\bullet$If $(X1, X2)$ be a bivariate Gaussian random variable with parameters $µ$ and $Σ_{x}$. Find the distribution of $X1 + X2$.
Hi all, for this question I'm not sure about the best way to approach it, as we are not given the $X1 \ \ and \ \ X2$ are independent, or of $X1 ,X2$ are gaussian. But as the Joint distribution is Gaussian, I know one of them must be true...any help would be appreicated.
Use the standard result: Joint distribution of two random variable is Gaussian implies that the marginal distributions of the random variables are also Gaussian.
In the above problem, let $\mu =(\mu_1,\mu_2)$ and $\Sigma_x=\begin{bmatrix}\Sigma_1 & \Sigma_{12} \\ \Sigma_{12} & \Sigma_2 \end{bmatrix}$.
Then the distribution of $(X_1+X_2)$ is Gaussian with mean $\mu_1+\mu_2$ and variance $\Sigma_1+\Sigma_2+2\Sigma_{12}$.