We have random vector (X,Y) with density $$g(x,y)=\frac{1}{\sqrt{2\pi}}\exp(-\frac{x^2-2xy+3y^2}{2})$$
I need to find if variables $X+Y$ and $X-Y$ are independent. One way to do it would be to read off density of $X$ and $Y$ and then find density of variables we need, its distributions functions and check it. I do not know if this is the best way to do it. But anyway I do not know how (not only in this particular case) how to find densities of single variables, knowing denisty of its vector.
$X$ and $Y$ are jointly normal. Then $X+Y$ and $X-Y$ are also jointly normal, and they are independent if and only if their covariance is nonzero. But $\text{Cov}(X+Y,X-Y) = \text{Var}(X) - \text{Var}(Y)$. Do $X$ and $Y$ have the same variance?