$X_1$, $X_2$ and $X_3$ are three independent random variables with the same density function $f_{Xi}(x)= e^{-x},\, i\in{\{1,2,3\}}$.
We also have \begin{align} Y_1 &= \frac{X_1}{X_1+X_2} \\ Y_2 &= \frac{X_1+X_2}{X_1+X_2+X_3} \\ Y_3 &= {X_1+X_2+X_3}. \end{align}
Using the Jacobian transformation. We find $f_{Y_1,Y_2Y_3}(y_1,y_2,y_3)= y_2y_3^2e^{-y_3},\, y_1,y_2,y_3>0$
I want to find the marginal densities of $Y_1,Y_2,Y_3$. For this, I want to use the joint density function $f_{Y_1,Y_2Y_3}(y_1,y_2,y_3).$
Example: $$f_{Y1}(y_1)=\int_0^?\int_0^? f_{Y_1,Y_2Y_3}(y_1,y_2,y_3) \,dy_2dy_3$$ But I can't find the limits of integration.
Thank you