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.
Best Answer
Take $X_1 \sim \mathcal N(0,1)$ and let $Z$ be a r.v. with $P(Z =-1)=P(Z=1)=1/2$ and $X_1, Z$ are independent. Denote $X_1'=Z\cdot X_1$, then $X_1' \sim \mathcal N(0,1)$. However $(X_1, X_1')$ is not Gaussian as $X_1+X_1'$ is not normal. Then $X_2$ should not Gaussian, perhaps $X_2$ will be Gaussian with some specific values of $\rho$. We have $Y_1$ is Gaussian, $Y_2$ is not Gaussian then $Y_1+Y_2$ is probably not Gaussian in general.