I describe the problem dimension $2$, but it could be generalized to $n$ dimensions.

So we have $X\_{1}, X\_{2}, X\_{3}$ three $iid$ random variables of continuous law $F(x,y)$ on $\Bbb R^2$. Let's denote by $S[X\_{1}, X\_{2}, X\_{3}]$ the simplex generated by those random variables. Given a point $(x,y)$, is it possible to write $P((x,y)\in S[X\_{1}, X\_{2}, X\_{3}])$ as a function $g$ of $F(x,y)$ ? For example, in dimension $1$, $P(x\in [X\_{1}, X\_{2}]) = 2F(x)(1-F(x))$.

Thank you