Suppose there are two correlated random variables $X_1$ and $X_2$ both are gamma distributed but having different shape and scale parameters with correlation coefficient $\rho$. What will be the distribution of Y? where $Y=(2X_1 X_2)/(X_1+X_2)$.

You say these variables are correlated. So the distribution of $Y$ will depend on the joint distribution of $X_1$ and $X_2$ (which knowing their individual distributions is insufficient to determine).
– Robin ChapmanAug 18 '10 at 6:47

@Robin Chapman, To find solution of this question, if more information required, i am providing link for joint distribution of $X_1$ and $X_2$. onlinelibrary.wiley.com/doi/10.1002/hyp.259/pdf
– user8576Aug 18 '10 at 8:37

@dikuve: Instead of referring the reader to a non freely accessible paper, you might want to address Robin's remark that the joint distribution of $(X_1,X_2)$ being not entirely determined by 1. the probability distribution of $X$, 2. the probability distribution of $Y$ and 3. the correlation coefficient $\rho$, your question has no answer (or, if these three parameters are enough to determine the probability distribution of $Y$, to explain why this is so).
– DidJan 9 '11 at 17:40