Support x ∈ R Variance Undefined | Mean Undefined MGF Undefined | |
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PDF 1 2 π i ∫ c − i ∞ c + i ∞ e s log s + x s d s {displaystyle { rac {1}{2pi i}}int _{c-iinfty }^{c+iinfty }!e^{slog s+xs},ds} | ||
In probability theory, the Landau distribution 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.
Contents
Definition
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
This distribution is a special case of the stable distribution with parameters α = 1, and β = 1.
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.