Landau distribution
Probability density function | |||
Parameters | — scale parameter | ||
---|---|---|---|
Support | |||
Mean | Undefined | ||
Variance | Undefined | ||
MGF | Undefined | ||
CF |
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.
Definition
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.
Properties
- Translation: If then .
- Scaling: If then .
- Sum: If and then .
These properties can all be derived from the characteristic function. Together they imply that the Landau distributions are closed under affine transformations.
Approximations
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
- The Landau distribution is a stable distribution with stability parameter and skewness parameter both equal to 1.
References
- ^ Landau, L. (1944). "On the energy loss of fast particles by ionization". J. Phys. (USSR). 8: 201.
- ^ 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.
- ^ Zolotarev, V.M. (1986). One-dimensional stable distributions. Providence, R.I.: American Mathematical Society. ISBN 0-8218-4519-5.
- ^ "LandauDistribution—Wolfram Language Documentation".
- ^ Behrens, S. E.; Melissinos, A.C. Univ. of Rochester Preprint UR-776 (1981).
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with finite support | |
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