 Solution

The inverse change of variables is $y_1=uv$, $y_2=(1-u)v$, on $0<u<1$, $v>0$. Its absolute Jacobian determinant isThe change of variables for probability densities therefore givesThis factorizes on a product domain into a unit uniform density in $u$ and a shape-two Gamma density in $v$. Thus $U$ and $V$ are independent, with $U\sim \mathrm{Uniform}(0,1)$ and $V\sim \mathrm{Gamma}(2,\lambda )$. This is the shape-$(1,1)$ case of beta-gamma independence.