# Mean and Variance of maximum of random variables

Given a set of random variables $x_1,x_2,...,x_n$, and we know their means and variances $(\mu_1,\sigma_1),(\mu_2,\sigma_2),...,(\mu_n,\sigma_n)$. How to compute mean and variance of the maximum distribution of $x_1,x_2,...,x_n$? We can assume $x_1,x_2,...,x_n$ are gaussian distributions.

• Maybe this is helpful: ocw.mit.edu/courses/civil-and-environmental-engineering/… Apr 28, 2018 at 8:55
• See stats.stackexchange.com/questions/18433/… and references therein, such as Wikipedia. Also again stats.SE for the Gaussian case. Apr 28, 2018 at 12:20
• @TobiasFritz -- those references are for the case of identically distributed Gaussian variables; the OP asks for the non-identical case, which seems quite a bit more complicated. Apr 28, 2018 at 12:22
• @CarloBeenakker: right, I wasn't paying proper attention. Thanks. Apr 28, 2018 at 12:28
• Do you just want an answer and don't care about having an elegant solution? Then do it by stochastic (Monte Carlo) simulation. You can incorporate whatever dependencies or distributions you want. Apr 28, 2018 at 12:36

For independently distributed $x_i$'s, each with cumulative distribution $$F_i(x_i)=\tfrac{1}{2}+\tfrac{1}{2}\operatorname{Erf}\,[(x_i-\mu_i)/(\sigma_i\sqrt 2],$$ the cumulative distribution of the maximum is given by $$P(\max_i \,x_i<X_{\max})=\prod_{i=1}^n P(x_i<X_{\max})=\prod_{i=1}^n F_i(X_{\max}).$$ For small $n$ you can now calculate moments of $X_{\rm max}$ by integration, $$E(X_{\max}^p)=\int_{-\infty}^\infty x^p\frac{d}{dx}\left(\prod_{i=1}^n F_i(x)\right)\,dx.$$ There is unlikely to be a closed-form answer for arbitrary $n$, in fact, even the $n=2$ integral seems problematic (Mathematica fails to evaluate it). If you take the $\mu_i$'s and $\sigma_i$'s to be the same, then progress can be made, for $n=2$ I find $$E(X_{\max})=\mu+\sigma/\sqrt\pi,\;\operatorname{Var}(X_{\max})=(1-1/\pi) \sigma^2.$$
Perhaps you are satisfied with a large-$n$ approximation. For identical $\mu_i$'s and $\sigma_i$'s it is given by the Fisher–Tippett–Gnedenko theorem, see for example this MSE posting. I have found one paper that generalizes this to arbitrary $\mu_i$'s and $\sigma_i$'s: On the distribution of the maximum of n independent normal random variables: iid and inid cases, but I have difficulty parsing their result (a rescaled Gumbel distribution).
There is more in that reference that I do not understand. They give the inequality $$\frac{1}{n}\sum_i\mu_i\leq E(X_{\rm max})\leq \frac{1}{n}\sum_i\mu_i+\frac{n-1}{n} \max_i\,\mu_i$$ which contradicts the $n=2$ result given above.
• I changed several instances of {\rm max} to \max. That affects spacing: with the latter the amount of space to the right and left depends on the context without any manual adjustments, and you formerly saw ${\rm max}_i$ rather than $\displaystyle\max_i$ near the bottom line. May 1, 2018 at 18:44