# expectation of maximum of iid random variables from normal distribution. [duplicate]

1) How to find expectation of max of random variables , i.e : $\mathbb{E}[max(x_1,x_2,\dots,x_n)]$ where $x$ are IID random variables from $\mathcal{N}(\mu,\sigma^2)$.

• I know that CDF is $F(x)^n$ and PDF is $nF(x)^{n-1}f(x)$. I have also seen simplifications for uniform and exponential distributions but not for Gaussian distribution.

2) In general, How to solve

$\int_{x=0}^{\infty} [nx F(x)^{n-1}f(x)] dx$

## marked as duplicate by NCh, José Carlos Santos, Namaste integration StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Mar 30 '18 at 14:42

• The normal distribution is, perhaps surprisingly, one of the more burdensome standard distributions to compute asymptotic maximum statistics for (but as the answer Math1000 linked to shows, it can be rescaled and recentered to converge to a Gumbel distribution, just like the exponential and for that matter, any continuous distribution with thinner-than-power-law tails and support everywhere). The upshot is the expectation of the maximum grows like $\sqrt{\log(n)}$ as the sample size increases. – spaceisdarkgreen Mar 30 '18 at 3:13