I have tried to use the method involving the Jacobian matrix but I am struggling to find the inverse of the function $f(x,y) = (\frac{x}{x+y}, \frac{y}{x+y})$.
We have that $X,Y \geq 0$.
2026-04-04 03:23:24.1775273004
What is the density of $(W, Z)$ where $W = \frac{X}{X+Y}$ and $ Z = \frac{Y}{X+Y}$?
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There is no inverse of $f$, for the two components of the result are related by $W+Z=1$. $f(1,2)=f(2,4)$, for example. It suffices to find the density of one of the two coordinates (e.g. by finding the density of $(X/(X+Y),Y)$) and then complementing to find the density of the other.
If $(W,Y)=(X/(X+Y),Y)$ then $YW/(1-W)=X$, i.e. $(X,Y)=(YW/(1-W),Y)$.