Half-Normal Distribution

Half-Normal Distribution

$\begin{figure}\begin{center}\BoxedEPSF{HalfNormalDistribution.epsf scaled 650}\end{center}\end{figure}$

A Normal Distribution with Mean 0 and Standard Deviation $1/\theta$ limited to the domain $[0, \infty)$ .

$\displaystyle P(x)$	$\textstyle =$	$\displaystyle {2\theta\over\pi} e^{-x^2\theta^2/\pi}$	(1)
$\displaystyle D(x)$	$\textstyle =$	$\displaystyle \mathop{\rm erf}\nolimits \left({tx\over\sqrt{\pi}}\right).$	(2)

The Moments are

$\displaystyle \mu_1$	$\textstyle =$	$\displaystyle {1\over t}$	(3)
$\displaystyle \mu_2$	$\textstyle =$	$\displaystyle {\pi\over 2t^2}$	(4)
$\displaystyle \mu_3$	$\textstyle =$	$\displaystyle {\pi\over t^3}$	(5)
$\displaystyle \mu_4$	$\textstyle =$	$\displaystyle {3\pi^2\over 4t^4},$	(6)

so the Mean, Variance, Skewness, and Kurtosis are

$\displaystyle \mu$	$\textstyle =$	$\displaystyle {1\over\theta}$	(7)
$\displaystyle \sigma^2$	$\textstyle \equiv$	$\displaystyle \mu_2-{\mu_1}^2={\pi-2\over 2t^2}$	(8)
$\displaystyle \gamma_1$	$\textstyle =$	$\displaystyle 2\sqrt{2\over\pi}$	(9)
$\displaystyle \gamma_2$	$\textstyle =$	$\displaystyle 0.$	(10)

See also Normal Distribution

© 1996-9 Eric W. Weisstein
1999-05-25