Product and quotient of correlated beta variables

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