Landau distribution
Probability density function


scale parameter

location parameter
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 "fat" tail, the moments of the distribution, such as mean or variance, are undefined. The distribution is a particular case of stable distribution.


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. In other words it is the Laplace transform of the function .

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:

Taking and we get the original form of above.


The approximation function for

These properties can all be derived from the characteristic function. Together they imply that the Landau distributions are closed under affine transformations.


In the "standard" case and , the pdf can be approximated[4] using Lindhard theory which says:

where is Euler's constant.

A similar approximation [5] of for and is:

Related distributions


  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. ^ "LandauDistribution—Wolfram Language Documentation".
  5. ^ Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).