Let X,YX, YX,Y be independent with X∼Gamma(2,1),Y∼Gamma(3,1)X \sim \text{Gamma}(2, 1), Y \sim \text{Gamma}(3, 1)X∼Gamma(2,1),Y∼Gamma(3,1). What is the PDF of Z=XX+YZ = \frac{X}{X+Y}Z=X+YX?
Beta(2,3)(2, 3)(2,3)
Beta(3,2)(3, 2)(3,2)
Beta(2,1)(2, 1)(2,1)
Beta(5,1)(5, 1)(5,1)