Landau distribution
In probability theory, the Landau distribution 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.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.
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, with characteristic function:
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 of for and :Related distributions
