I was given the density of a 2-dimensional normal distribution and calculated the densities of $X_1, X_2$. My calculation resultet in both $X_1, X_2$ being distributet $N(0,\frac{3}{8})$ but with covariance $\mathrm{Cov}(X_1, X_2) = - \frac{1}{8}$. Therefore they can not be independend, thus I can not use the convolution to calculate the density of $X_1 + X_2$. How can I continue?
Also I want to calculate the joint density of $X_1 + X_2$ and $X_1 - X_2$.
Edit: I now get the following solutions $$X_1 + X_2 \sim N(0, \frac{1}{2})$$ $$X_1 - X_2 \sim N(0, -\frac{1}{4})$$ $$(X_1 + X_2, X_1 - X_2) \sim N(\begin{pmatrix}0 \\ 0\end{pmatrix}, \begin{pmatrix}\frac{1}{2} & \frac{3}{2}\\ \frac{3}{2} & -\frac{1}{4}\end{pmatrix})$$ I would appreciate it, if someone could check those results.
Let $Y_1=X_1$ and $Y_2=X_1+3X_2$. Then $Y_1,Y_2$ also have a joint normal distribution and $cov(Y_1,Y_2)=EX_1^{2}+3EX_1X_2=0$. This makes $Y_1$ and $Y_2$ independent normal variables. You have to find the joint density of $X_1+X_2=\frac {Y_2+2Y_1} 3$ and $X_1-X_2=\frac {4Y_1-Y_2} 3$. Find the joint density of $(Y_1,Y_2)$ using independence and apply Jacobian formula to get the joint density of $\frac {Y_2+3Y_1} 3$ and $\frac {4Y_1-Y_2} 3$.