Given two random variables X and Y that are independent and

Given two random variables X and Y that are independent and uniformly distributed as U(0,1): Find the joint pdf f_U,V of random variables U and V defined as: Sketch the support of f_U,V in the (u, v) plane. Remember support of a function is the subset of its domain for which the function takes on nonzero values.

Solution

f(x)=1/(b-a), a<x<b, and 0 otherwise. Given that X~U(0,2) & Y~U(0,1), and given U=1/2(X+Y), V=1/2(X-Y) ,then jacobian transformation is -2, then f(U,V)=f(x,y)|J|=2, 0<U<1 & -1<V<0.