Landau distribution

In probability theory, the Landau distribution[1] is a probability distribution named after Lev Landau. Because of the distribution's "fat" tail, the moments of the distribution, like mean or variance, are undefined. The distribution is a particular case of stable distribution.

Landau distribution
Probability density function

scale parameter

location parameter
Mean Undefined
Variance Undefined
MGF Undefined


The probability density function, as written originally by Landau, is defined by the complex integral:

where a is an arbitrary positive real number, meaning that the integration path can be any parallel to the imaginary axis, intersecting the real positive semi-axis, and refers to the natural logarithm.

The following real integral is equivalent to the above:

The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters and [2], with characteristic function[3]:

where and , which yields a density function:

Let us note that the original form of is obtained for and , while the following is an approximation[4] of for and :

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


  1. Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201.
  2. 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.
  3. Zolotarev, V.M. (1986). One-dimensional stable distributions. Providence, R.I.: American Mathematical Society. ISBN 0-8218-4519-5.
  4. Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).
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