Landau distribution

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

In probability theory, the Landau distribution^[1] 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.

Definition

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

p(x)={\frac {1}{2\pi i}}\int _{{c-i\infty }}^{{c+i\infty }}\!e^{{s\log s+xs}}\,ds,

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,

p(x)={\frac {1}{\pi }}\int _{0}^{\infty }\!e^{{-t\log t-xt}}\sin(\pi t)\,dt.

The full family of Landau distributions is obtained by extending the standard distribution to a location-scale family. This distribution can be approximated by ^[2]^[3]

p(x)={\frac {1}{{\sqrt {2\pi }}}}\exp \left\{-{\frac {1}{2}}(x+e^{{-x}})\right\}.

This distribution is a special case of the stable distribution with parameters α = 1, and β = 1.^[4]

The characteristic function may be expressed as:

\varphi (t;\mu ,c)=\exp \!{\Big [}\;it\mu -|c\,t|(1+{\tfrac {2i}{\pi }}\log(|t|)){\Big ]}.

where μ and c are real, which yields a Landau distribution shifted by μ and scaled by c.^[5]

Related distributions

If $X\sim {\textrm {Landau}}(\mu ,c)\,$ then $X+m\sim {\textrm {Landau}}(\mu +m,c)\,$
The Landau distribution is a stable distribution with the stable distribution stability parameter α and skewness parameter β both equal to 1

References

↑ Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201.
↑ Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).
↑ "Interaction of Charged Particles". Retrieved 14 April 2014.
↑ Gentle, James E. (2003). Random Number Generation and Monte Carlo Methods. Statistics and Computing (2nd ed.). New York, NY: Springer. p. 196. doi:10.1007/b97336. ISBN 978-0-387-00178-4.
↑ Meroli, S. (2011). "Energy loss measurement for charged particles in very thin silicon layers". JINST. 6: 6013.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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