= Expected maximum of two absolute standard normal variables
{title2=$\mathbb E\max(|N_1|,|N_2|)=2/\sqrt\pi$}
For two <independent> <standard normal random variables>, the maximum of their absolute values has distribution function $(2\Phi(x)-1)^2$ for $x\geq0$. Differentiation and <integration by parts>, using $\varphi'(x)=-x\varphi(x)$, give its expectation as $8\int_0^\infty\varphi(x)^2\,dx=2/\sqrt\pi$. This exact value can detect missing factors in two-sided Gaussian-maximum estimates.
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