Landau distribution

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Landau distribution
Probability density function
Landau distribution

c ∈ (0, ∞) — scale parameter

μ ∈ (−∞, ∞) — location parameter
Support xR
Mean Undefined
Variance Undefined
MGF Undefined

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.


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 [2][3]

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:

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

Related distributions[edit]

  • If then
  • The Landau distribution is a stable distribution with the stable distribution stability parameter α and skewness parameter β both equal to 1


  1. ^ Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201. 
  2. ^ Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981). 
  3. ^ "Interaction of Charged Particles". Retrieved 14 April 2014. 
  4. ^ 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. 
  5. ^ Meroli, S. (2011). "Energy loss measurement for charged particles in very thin silicon layers". JINST. 6: 6013.