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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,

$\displaystyle P(x)$ $\textstyle =$ $\displaystyle {2\theta\over\pi} e^{-x^2\theta^2/\pi}$ (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