The question I am attempting is the one shown below: 
It is easy enough to find $F_U , F_V $ from definitions, and I have shown that $U \sim \epsilon (2) $ by subbing in to the expression for $F_U$. However I am unsure how to show $V-U, U$ are independent.
Usually to show this kind of thing, I would consider a map from some original r.v.s to the new ones, and the joint density function for the new r.v.s will be the previous with Jacobian factor ; If this is separable into the 2 new r.v.s they must be independent.
But here I don't know the joint distribution for $U,V$, and considering map from $X,Y$ to $V-U, U$ seems fiddly since we bring in moduli.
Should I be able to find the distribution of $V-U$ easily?
If I can find this, then the last part can be done by convolution of $U, V-U$ distributions.
Any help on how to solve this would be appreciated.
