World Library  
Flag as Inappropriate
Email this Article

Half-normal distribution


Half-normal distribution

Half-normal distribution
Parameters σ2 > 0 — (scale)
Support x ∈ [0,\infty)
PDF f(x; \sigma) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp \left( -\frac{x^2}{2\sigma^2} \right) \quad x>0
CDF F(x; \sigma) = \mbox{erf}\left(\frac{x}{\sigma\sqrt{2}}\right)
Quantile Q(F;\sigma)=\sigma\sqrt{2}\mbox{erf}^{-1}(F)
Mean \frac{\sigma\sqrt{2}}{\sqrt{\pi}}
Median \sigma\sqrt{2}\mbox{erf}^{-1}(1/2)
Variance \sigma^2\left(1 - \frac{2}{\pi}\right)
Entropy \frac{1}{2} \log \left( \frac{ \pi \sigma^2 }{2} \right) + \frac{1}{2}

The half-normal distribution is a special case of the folded normal distribution.

Let X follow an ordinary normal distribution, N(0,\sigma^2), then Y=|X| follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.

Using the \sigma parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by

f_Y(y; \sigma) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp \left( -\frac{y^2}{2\sigma^2} \right) \quad y>0,

Where E[Y] = \mu = \frac{\sigma\sqrt{2}}{\sqrt{\pi}}.

Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac{\sqrt{\pi}}{\sigma\sqrt{2}}, the probability density function is given by

f_Y(y; \theta) = \frac{2\theta}{\pi}\exp \left( -\frac{y^2\theta^2}{\pi} \right) \quad y>0,

where E[Y] = \mu = \frac{1}{\theta}.

The cumulative distribution function (CDF) is given by

F_Y(y; \sigma) = \int_0^y \frac{1}{\sigma}\sqrt{\frac{2}{\pi}} \, \exp \left( -\frac{x^2}{2\sigma^2} \right)\, dx

Using the change-of-variables z = x/(\sqrt{2}\sigma), the CDF can be written as

F_Y(y; \sigma) = \frac{2}{\sqrt{\pi}} \,\int_0^{y/(\sqrt{2}\sigma)}\exp \left(-z^2\right)dz = \mbox{erf}\left(\frac{y}{\sqrt{2}\sigma}\right),

where erf(x) is the error function, a standard function in many mathematical software packages.

The quantile function (or inverse CDF) is written:


where 0\le F \le 1 and \mbox{erf}^{-1}() is the inverse error function

The expectation is then given by

E(Y) = \sigma \sqrt{2/\pi},

The variance is given by

\operatorname{Var}(Y) = \sigma^2\left(1 - \frac{2}{\pi}\right).

Since this is proportional to the variance σ2 of X, σ can be seen as a scale parameter of the new distribution.

The entropy of the half-normal distribution is exactly one bit less the entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus,

H(Y) = \frac{1}{2} \log \left( \frac{ \pi \sigma^2 }{2} \right) + \frac{1}{2}

Differential equation

\left\{\sigma ^2 f'(x)+x f(x)=0,f(1)=\frac{\sqrt{\frac{2}{\pi }} e^{-\frac{1}{2 \sigma ^2}}}{\sigma }\right\}
\left\{\pi f'(x)+2 \theta ^2 x f(x)=0,f(1)=\frac{2 e^{-\frac{\theta ^2}{\pi }} \theta }{\pi }\right\}


  • Parameter estimation 1
  • Related distributions 2
  • External links 3
  • References 4

Parameter estimation

Given numbers \{x_i\}_{i=1}^n drawn from a half-normal distribution, the unknown parameter \sigma of that distribution can be estimated by the method of maximum likelihood, giving

\hat \sigma = \sqrt{\frac 1 n \sum_{i=1}^n x_i^2}

Related distributions

External links

(note that MathWorld uses the parameter \theta = \frac{1}{\sigma}\sqrt {\pi/2} )


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.