Let $X \sim \cal{N}(1, 3)$ and $Y \sim \cal{N}(2, 3)$ random variables. Suppose that $\operatorname{cov}(X, Y) = 1$. Consider transformation of these random variables: $U = X + Y$ and $V = X − Y$.
I need to find the joint density function of $U$ and $V$.
How to tackle this problem. What can I get from $\operatorname{cov}(X, Y) = 1$? I've tried a few ways but ended up with nothing.
This approach involves a reference to PCA.
Say, $X$ & $Y$ are standardized and the correlation between them is $\rho$. In terms of correlation matrix, it can be written as $$ \begin{pmatrix} 1 & \rho \\ \rho & 1 \\ \end{pmatrix} $$
If you find the eigenvectors of this correlation matrix, you will see that $\frac{1}{\sqrt{2}}(1,1)$ and $\frac{1}{\sqrt{2}}(1,-1)$ are the eigenvectors. Essentially, in these two directions, we can find the maximum and minimum variance - Reason behind why they are called Principal Components (PC).
Also, an important property of these PCs is that they are orthogonal to each other - an increase/decrease along one direction is not going to affect in the other direction.
Now, coming to your original problem, since $U$ and $V$ are along the directions of PCs, we can assume that they are independent. Therefore, you can simply multiply the marginal distributions to get the joint distribution.