Sum of correlated normal random variables Suppose I have two correlated random variables, that were generated in the following way:
\begin{align*}
X_1 &\sim \mathcal{N}(0,1)\\
X_1' &\sim \mathcal{N}(0,1)\\
X_2 &= \rho X_1+\sqrt{1-\rho^2}\cdot X_1'\\
Y_1 &= \mu_1+\sigma_1 X_1\\
Y_2 &= \mu_2+\sigma_2 X_2.
\end{align*}
Now, is it true that $Y_1+Y_2$ (or, more generally $\alpha_1 Y_1+\alpha_2Y_2$) normally distributed? (I can easily calculate the mean and the variance of
$\alpha_1 Y_1+\alpha_2Y_2$, but I am not sure about the distribution...)
EDIT: just to clarify, $X_1$ and $X_1'$ are independent.

$\alpha_1 Y_1 + \alpha_2 Y_2$ is a linear combination of $X_1$ and $X_1^\prime$ - that is $\alpha_1 Y_1 + \alpha_2 Y_2 = \beta X_1 + \beta^\prime X_1^\prime$ for some $\beta, \beta^\prime$ that are a bit of a pain to calculate.  Linear combinations of independent normal random variables are normal; there are several proofs of this (nontrivial, but well-known) fact.   So the answer to your question is yes.
