Let's say we have 50 independent random variables $X_i$ with the same distribution. Let's also say that $E(X_i) = \mu$ and $Var(X_i) = \sigma^2 > 0$ are known values. Is it possible to determine a (non trivial) lower bound for $E(\max X_i)$?
I've found some questions here concerning uniform distributions, but I was wondering if we could say something about this question when distribution of $X_i$s is unknown (other than ($E(\max X_i)\geq \mu)$).