For question (ii) I don't understand the solution, which is:
What I don't understand:
Why would their joint density be constant on the square $[-1,1]$ if these variables were to be independent? And why is it not constant if their not independent? I don't understand anything to be honest in this solution to (ii), only the part which states that their densities are symmetric, which are equal to $\frac{1}{2}$ for $|x|\lt 1$. If someone could tell me where I could read up some theory for this or explain elaborately, it would be very helpful. Thanks
It has been established in $(a)$ that $X,Y\sim U[-1,1]$. If $X,Y$ are independent, $f_{XY}(x,y)=f_X(x)f_Y(y)=1/4$ for $-1\le x,y\le1$, which is not the case.