I have $n$ i.i.d. random variables, $X_1,..., X_n$ which follow some arbitrary distribution. Based on experiments in Python with various distributions, it seems that $\mathbb{E}(\max(X_1,...,X_n))$ is a linear (or seemingly close to linear) function of $\mathbb{E}(X_i)$. It is indeed linear for some examples where it is possible to get a closed form solution for $\mathbb{E}(\max(X_1,...,X_n))$ or a good approximation.
Expected value of $\max\{X_1,\ldots,X_n\}$ where $X_i$ are iid uniform.
Expectation of the maximum of i.i.d. geometric random variables
I wonder if this is the case more generally? Is there some way to prove it?