Question: Let X and Y be two continuous random variables with joint probability density function
$$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 :\ \ \ \ \ \ \ \ \ \text{otherwise}\end{cases}$$
Find the joint probability density function of $U=X+Y$ and $V=X-Y$ And then find the covariance matrix of the distribution
My attempt:
For the first part I think I have managed to work out the correct answer.I got,
$$f(u,v)=\begin{cases}\frac{1}{4} & \text{if} \ \lvert u \rvert \le 1, \lvert v \rvert \le 1 & \\ 0 :\ \ \ \ \ \ \ \ \text{otherwise}\end{cases}$$
For the covariance part, I got $Cov(U,V)=0$ and $Var(U) = \frac{1}{3}$ and $Var(V) = \frac{1}{3}$ However this doesn't feel right.
Is this correct? Also can we tell from the covariance matrix, whether or not this is independent?