Let X and Y be independent Gamma(, 1) and Gamma(, 1) random variables. The shape parameters are

Let X and Y be independent Gamma(α, 1) and Gamma(β, 1) random variables. The shape parameters are α and β respectively and the rate parameters are 1. 

Let U = X + Y and V = X/(X + Y ). 

(a) Determine the joint probability density function (PDF) f_X,Y (x, y) of (X, Y ).

(b) Determine the joint PDF of (U, V ).

(c) Determine the marginal PDF of V , that is f_V (v). Is V a Gamma random variable? If not, then what distribution does it have?

(d) Are U and V independent? Explain