Probability density function [math]\displaystyle{ \mu=0,\; c=\pi/2 }[/math] | |||
Parameters |
[math]\displaystyle{ c \in(0,\infty) }[/math] — scale parameter | ||
---|---|---|---|
Support | [math]\displaystyle{ \mathbb{R} }[/math] | ||
[math]\displaystyle{ \frac{1}{\pi c}\int_0^\infty e^{-t}\cos\left(t\left(\frac{x-\mu}{c}\right) + \frac{2t}{\pi}\log\left(\frac{t}{c}\right)\right)\, dt }[/math] | |||
Mean | Undefined | ||
Variance | Undefined | ||
MGF | Undefined | ||
CF | [math]\displaystyle{ \exp\left(it\mu -\frac{2ict}{\pi}\log|t| - c|t|\right) }[/math] |
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, like mean or variance, are undefined. The distribution is a particular case of stable distribution.
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 [math]\displaystyle{ \log }[/math] refers to the natural logarithm. In other words it is the Laplace transform of the function [math]\displaystyle{ s^s }[/math].
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 [math]\displaystyle{ \alpha=1 }[/math] and [math]\displaystyle{ \beta=1 }[/math],[2] with characteristic function:[3]
where [math]\displaystyle{ c\in(0,\infty) }[/math] and [math]\displaystyle{ \mu\in(-\infty,\infty) }[/math], which yields a density function:
Taking [math]\displaystyle{ \mu=0 }[/math] and [math]\displaystyle{ c=\frac{\pi}{2} }[/math] we get the original form of [math]\displaystyle{ p(x) }[/math] above.
These properties can all be derived from the characteristic function. Together they imply that the Landau distributions are closed under affine transformations.
In the "standard" case [math]\displaystyle{ \mu=0 }[/math] and [math]\displaystyle{ c=\pi/2 }[/math], the pdf can be approximated[4] using Lindhard theory which says:
where [math]\displaystyle{ \gamma }[/math] is Euler's constant.
A similar approximation [5] of [math]\displaystyle{ p(x;\mu,c) }[/math] for [math]\displaystyle{ \mu=0 }[/math] and [math]\displaystyle{ c=1 }[/math] is:
Original source: https://en.wikipedia.org/wiki/Landau distribution.
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