Let's say I have $N$ independent uniformly distributed random variables $U(a_i, b_i)$ and I'm interested in $$P(U_1 - U_2\cdots - U_N < 0) \text{ .}$$
How would I calculate something like that? I'm assuming expression above means that $U_1$ ends up being smaller than all of the other variables. There are trivial cases where $b_1 < a_i$ but for the overlapping distributions it looks hard to calculate.
I do know that a difference $U_i - U_j$ is a triangular distribution, so I could just calculate the area of the triangles/squares but with more variables in the difference the calculation gets too complicated.