Suppose we have a random variables $R_1$ and $R_2$ such that they are uniformly distributed on the triangle $\{(x,y):x,y>0, x+y<1.\}$ Or explicitly, they are uniform on a triangle formed by the $x$-axis, $y$-axis and $x+y=1.$ Now, suppose I have another random variable $S\sim\Gamma(3, \lambda)$ and this is independent to $R_1, R_2$. In other words the density is given as $\frac{1}{2}\lambda^3s^2e^{-\lambda s}.$ What procedures can I follow to find the joint density of $R_1S$ and $R_2S$?
I did calculate the joint density of $R_1, R_2$ and subsequently I calculated the marginal distribution of $R_1$ and $R_2$. I could calculate the joint density of $R_1, S$ and the joint densities of $R_2, S$. Then I used these densities to find the joint density of $R_1S, S$ and $R_2S, S.$ Now I am stuck, can I deduce the joint density of $R_1S, R_2S$ from here?
Many thanks in advance!