Moyal Distribution#

Univariate, Continuous, Asymmetric, Unbounded, Light-tailed

The Moyal distribution is a continuous probability distribution that was proposed by the physicist J. E. Moyal in 1955 as an approximation to the Landau distribution. The Moyal distribution is characterized by two parameters: the location parameter \(\mu\) and the scale parameter \(\sigma\).

The Moyal distribution is used in high-energy physics to model the energy loss, and the number of ion pairs produced, by ionization for fast charged particles.

Key properties and parameters#

Support

\(x \in (-\infty, \infty)\)

Mean

\(\mu + \sigma\left(\gamma + \log 2\right)\), where \(\gamma\) is the Euler-Mascheroni constant

Variance

\(\frac{\pi^{2}}{2}\sigma^{2}\)

Parameters:

  • \(\mu\) (loc): The location parameter.

  • \(\sigma\) (scale): The scale parameter.

Probability Density Function (PDF)#

\[ f(x|\mu, \sigma) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}\left(z + e^{-z}\right)} \]

where \(z = \frac{x - \mu}{\sigma}\).

../../_images/7d75101895639fc4f4fb45bb7bd8fd33054d0cead22a889b21bb7ea33878dcd1.png

Cumulative Distribution Function (CDF)#

\[ F(x; \mu, \sigma) = 1 - \text{erf}\left( \frac{e^{-z}}{\sqrt{2}} \right) \]

where erf is the error function and \(z = \frac{x - \mu}{\sigma}\).

../../_images/4046a4c8a462a9cf97949b57063e4fbdb195111b3f301f15f4cbc03909a0fb53.png

See also

Related Distributions:

  • Gamma - The Moyal distribution is a transformation of the Gamma distribution.

  • LogNormal - The LogNormal distribution is also used sometimes as an approximation to the Landau distribution. It is suitable for modeling positive and right-skewed data.

References#