Suppose $A_i$ are independent uniformly distributed random variables on possibly different intervals: $A_i \sim U(0,b_i)$. I am interested in determining the probability that: \begin{equation} \left( \sum_{i=n}^{N}A_i \leq C_1 \quad \cap \quad \sum_{j=m}^{M}A_j \leq C_2 \right) \end{equation} where $m>n$ and $M>N$, but $m\leq N$. That is, the intervals $[n,N]$ and $[m,M]$ overlap. This means that some of the random variables appear in both sums, so the LHS and RHS are not independent.
I am struggling to understand how to relate the two sums to be able to determine their joint probability. Can anyone help me out?