Given $f_{X_1,X_2,X_3}(x_1,x_2,x_3)=e^{-(x_1+x_2+x_3)}$ for $x_1>0,x_2>0,x_3>0$ (zero elsewhere). Find the joint density of $Y_1=X_1+X_2+X_3$, $Y_2=X_2$ and $Y_3=X_3$. Then find the marginal density of $Y_1$.

I know how to find the joint density of the $Y$'s. I calculated the Jacobian and have $|J|=1$. So then the joint density is just $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3)=e^{-y_1}$. I know that $y_2>0,y_3>0$ and $y_1>y_2+y_3$.

I know that in order to find $f_{Y_1}(y_1)$ that I must integrate $y_2$ and $y_3$ out of $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3)=e^{-y_1}$, but I am unsure of how to figure out the bounds of integration.



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