What is the distribution of the sum of two normally distributed random variables that are NOT independent?

Question

Let $X \sim \mathcal{N}(\mu,\Sigma)$ be a random vector with two components, say $X_1$ and $X_2$. Then what is the distribution of the sum $X_1 + X_2$ in general case, assuming $X_1$ and $X_2$ can be dependent or independent? Please show me the calculation in details. Thanks!

Answer 1

hint

Let's say they're both standard $N(0,1)$ with correlation $\rho,$ i.e. $\Sigma = \begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}.$ In this case we can write an equality in distribution $$X_2 = \rho X_1 + \sqrt{1-\rho^2} Z$$ where $Z$ is a standard normal independent from $X_1.$

(If you haven't seen this before, the idea is this: $X_1$ and $X_2$ are linear combinations of the independent $X_1$ and $Z$ and thus are bivariate normal. Since bivariate normal is fully characterized by its mean vector and covariance matrix, it remains to calculate these for $(X_1,X_2)$ and confirm they are what you want.)

Then we have $$X_1+X_2 = (1+\rho)X_1 + \sqrt{1-\rho^2}Z \sim N(0,(1+\rho)^2+(1-\rho^2)).$$

Will leave it to you to generalize this.