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.
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.
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.
- If then
- The Landau distribution is a stable distribution
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