Let $x_1, x_2, x_3$ be iid random variables, and $$ y_1=x_1+x_2+x_3,\qquad y_2=x_1+x_2 $$
How can we derive the joint distribution of $y_1$ and $y_2$. In this case a one-to-one mapping does not exist so usual method involving jacobian cannot be used.