You right, the Jacobian is abs(det[y2 y1;1-y2 -y1 ])=y1 so you get f(y1,y2)=y1*exp(-y1)

Since the event{X₁+X₂=y₁,X₁/(X1+X2)=y₂}is equivalent with the event{X₂=y₁(1-y&#8322,X₁=y₂y₁}we obtain for the densityf(y1,y2)=exp(-y1*(1-y2))*exp(-y1*y2) =exp(-y1)