Let $X_i$, $1\le i\le t$, be $t$ independent random variables with Binomial distribution $B(n,\frac1t)$.
I would like to find the distribution of $X_{Max}=\max_{i=1}^t(X_i)$
Note that this is the specific case where the probability of success is equal to the reciprocal of the number of variables I am taking the maximum over.
I expect that the answer will be too complex to use directly, so an approximation would be good to have. In fact just having an approximate formula for $\mathbb{E}(X_{Max})$ and $\mathbb{Var}(X_{Max})$ would be sufficient, and I am most interested in the case where $n\gg t\gg1$.