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For independently distributed $x_i$'s, each with cumulative distribution $F_i(x_i)$, the cumulative distribution of the maximum is given by $$P({\rm max}_i \,x_i<X_{\rm max})=\prod_{i=1}^n P(x_i<X_{\rm max})=\prod_{i=1}^n F_i(X_{\rm max}).$$ For small $n$ you can now calculate moments of $X_{\rm max}$ by integration, $$E(X_{\rm max}^p)=\int_{-\infty}^\infty x^p\frac{d}{dx}\left(\prod_{i=1}^n F_i(x)\right)\,dx.$$ There is unlikely to be a closed-form answer for arbitrary $n$, but large-$n$ approximations do exist, along the lines of this MSE posting.