Statistics example 5 Joint distribution of random variables X and Y is 
$$f (x, y) = e^{-(x+y)} $$ for $$ 0<x,y<∞$$  and 0 otherwise. 
Determine the distribution densities of random variables U = X + Y and V = X/X + Y.

There are several ways to do this: 
a) CDF method: find $P(X+Y < u)$ by integration
b) you can notice that $X,Y$ are independent Exponential, and you probably studied what the distribution of their sum is... 
c) Jacobian method: see e.g. here 
http://web.eecs.umich.edu/~aey/eecs501/lectures/jacobian.pdf
