  Motivated by the celebrated Beck-Fiala conjecture, we consider the random
setting where there are $n$ elements and $m$ sets and each element lies in $t$
randomly chosen sets. In this setting, Ezra and Lovett showed an $O((t \log
t)^{1/2})$ discrepancy bound in the regime when $n \leq m$ and an $O(1)$ bound
when $n \gg m^t$.
  In this paper, we give a tight $O(\sqrt{t})$ bound for the entire range of
$n$ and $m$, under a mild assumption that $t = \Omega (\log \log m)^2$. The
result is based on two steps. First, applying the partial coloring method to
the case when $n = m \log^{O(1)} m$ and using the properties of the random set
system we show that the overall discrepancy incurred is at most $O(\sqrt{t})$.
Second, we reduce the general case to that of $n \leq m \log^{O(1)}m$ using LP
duality and a careful counting argument.
