Landau distribution

Landau distribution

Probability density function
Parameters	c ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R
PDF	${\displaystyle {\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\!e^{s\log s+xs}\,ds}$
Mean	Undefined
Variance	Undefined
MGF	Undefined
CF	${\displaystyle \exp \!{\Big [}\;it\mu -|c\,t|(1+{\tfrac {2i}{\pi }}\log(|t|)){\Big ]}}$

c ∈ (0, ∞) — scale parameter

In probability theory, the Landau distribution is a probability distribution named after Lev Landau. Because of the distribution's long tail, the moments of the distribution, like mean or variance, are undefined. The distribution is a special case of the stable distribution.

The probability density function of a standard version of the Landau distribution is defined by the complex integral

where c is any positive real number, and log refers to the logarithm base e, the natural logarithm. The result does not change if c changes. For numerical purposes it is more convenient to use the following equivalent form of the integral,

The full family of Landau distributions is obtained by extending the standard distribution to a location-scale family. This distribution can be approximated by

This distribution is a special case of the stable distribution with parameters α = 1, and β = 1.

The characteristic function may be expressed as:

where μ and c are real, which yields a Landau distribution shifted by μ and scaled by c.