Product and quotient of correlated beta variables

Let U , V , W be independent random variables having a standard gamma distribution with respective shape parameters a , b , c , and define X = U / ( U + W ) , Y = V / ( V + W ) . Clearly, X and Y are correlated each having a beta distribution, X ∼ B ( a , c ) and Y ∼ B ( b , c ) . In this article we derive probability density functions of X Y , X / Y and X / ( X + Y ) .