Sommelier of bell-shaped functions

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Q0. Functions are scaled so that \( f(0) = -f''(0) = 1 \).

		0	1	2	3	4
$$ \exp \left( -\frac{1}{2} x^2 \right) $$	Gauss dist.
$$ \frac{1}{\cosh x} $$	Hyperbolic secant
$$ \frac{1}{\cosh^2 (x / \sqrt{2})} = \tanh' \frac{1}{\sqrt{2}} x $$	Derivative of Fermi dist.
$$ \frac{1}{2} + \frac{1}{2} \cos \left( \frac{\pi}{2} x \right) \ \mathrm{if}\ \|x\| \lt 2, \ 0 \ \mathrm{otherwise}. $$	Hann window
$$ \frac{1}{1 + x^2 / 2} = \arctan' \frac{1}{\sqrt{2}} x $$	Cauchy dist.

Score: .

Q1. Functions are scaled so that \( \int dx\, f(x) = \int dx\, x^2 f(x) = 1 \).

		0	1	2	3	4
$$ \frac{1}{\sqrt{2 \pi}} \exp \left( -\frac{1}{2} x^2 \right) $$	Gauss dist.
$$ \frac{1}{2} \frac{1}{\cosh(\pi x / 2)} $$	Hyperbolic secant
$$ \frac{a}{2} \frac{1}{\cosh^2 a x} = \frac{a}{2} \tanh' a x,\ a = \frac{\pi}{2 \sqrt{3}} $$	Derivative of Fermi dist.
$$ \frac{a}{2 \pi} (1 + \cos a x) \ \mathrm{if}\ \|x\| \lt \frac{\pi}{a}, \ 0 \ \mathrm{otherwise},\ a = \sqrt{\frac{\pi^2 - 6}{3}} $$	Hann window
$$ \frac{2}{\pi} \frac{1}{(1 + x^2)^2} $$	ν = 3 Student's t-dist.

Score: .

(c) Yasuhiro Fujii <y-fujii at mimosa-pudica.net>