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. 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 , with characteristic function: 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 distribution is closed under affine transformations. Approximations In the "standard" case and , the pdf can be approximated using Lindhard theory which says: where is Euler's constant. A similar approximation of for and is: Related distributions The Landau distribution is a stable distribution with stability parameter and skewness parameter both equal to 1. References Continuous distributions Probability distributions with non-finite variance Power laws Stable distributions Lev Landau