I am sorry if there is no simple answer to this or the answer is completely obvious but I am approaching my wits end here. Probability isn't my forte, nor am I even a mathematician.
I am essentially looking at a matrix of Random variables $X_{ij}$ where each $X_{ij}$ is uniformly distributed $X_{ij} \sim U([0,1])$ and would then want to look at the sums of column variables.
In particular, I am interested in the distribution of a specific column sum minus the maximum of all other column sums. $$ \Bbb P\left(\sum_{i=1}^nX_{ij}-\left(\max_{k\neq j}{\sum_{i=1}^nX_{ik}}\right)<x\right) $$
Any comments or help would be appreciated, although doubts that there is an easy answer to this are starting to creep up on me.