I have problem when I try to evaluate $P(X_1+X_2<c \vert X_1<c)$, where $X_1$ and $X_2$ are independent and identically distributed.
More generally, I want to find $$P(\sum_{k=1}^n X_k<C \vert X_1<C, X_1+X_2<C,\dots, \sum_{k=1}^{n-1} X_k<C),$$
where all the $X_i$'s are i.i.d.
I was trying to do the integration over the joint PDF, but in my case, it may involve high dimensional integration (maybe 12 dimension), and the boundary conditions also look messy to me. So I'm stuck and don't know where to start.