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Let $X \sim Bin(n,p)$. Let $\{X_i\}$ be $n$ iid copies of $X$. Let $Z = \max{X_i}$. I want to put upper bounds on $E[Z]$. The variance of $X$ is $np(1-p)$. Following the analogy from the sub-Gaussian case I would say that $$E[Z] \leq np + O(\sqrt{np(1-p)\log(n)})$$ However I have not been able to prove the above statement, partly because for general $p$ I have not been able to establish sub-Gaussian behavior of the random variable $Bin(n,p)$ with the parameter $np(1-p)$.

Thanks in advance.

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