What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables I tried to derive it : $$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^k$$ $$E[max] = \sum_{C = 1}^n C(pr[max \leq C] - pr[max \leq C - 1]) = \sum_{C = 1}^n c((\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^k - (\sum_{i = 0}^{C - 1} {n \choose i}p^i(1 - p)^i)^k)$$ $$= n(\sum_{i = 0}^n {n \choose i}p^i(1 - p)^i)^k - \sum_{C = 0}^{n - 1}(\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^k = $$ $$n - \sum_{C = 0}^{n - 1}(\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^k$$
But my brain is not braining and Im not able to get to tight and nice lower and upper bounds for the very general case of expected max of k IID random variables with distribution binomial(n, p), I did an initial search engine search and I could not find anything useful I'd appreciate it if you could let me know of trivial or famous bounds that Im missing here, thank you so much.
