Editing Landau distribution

{{Short description|Probability distribution}}
{{Probability distribution
  | name       = Landau distribution
  | type       = density
  | pdf_image  = [[File:Landau Distribution PDF.svg|350px]]<br /><small><math>\mu=0,\; c=\pi/2</math></small>
  | support    = <math>\mathbb{R}</math>
  | parameters = <math>c \in(0,\infty)</math> — [[scale parameter]] <br>
                 <math>\mu\in(-\infty,\infty)</math> — [[location parameter]]
  | char       = <math>\exp\left(it\mu -\frac{2ict}{\pi}\log|t| - c|t|\right)</math>
  | mean       = Undefined
  | variance   = Undefined
  | mgf        = Undefined
  | pdf        = <math> \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>
  }}

In [[probability theory]], the '''Landau distribution'''<ref>{{ cite journal | last = Landau | first = L. | title = On the energy loss of fast particles by ionization | journal = J. Phys. (USSR) |url=http://e-heritage.ru/Book/10093344 | volume = 8 | page = 201 | date = 1944 }}</ref> is a [[probability distribution]] named after [[Lev Landau]].
Because of the distribution's "fat" tail, the [[Moment (mathematics)|moments]] of the distribution, such as 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 number|complex]] [[integral]]:

:<math>p(x) = \frac{1}{2 \pi i} \int_{a-i\infty}^{a+i\infty} e^{s \log(s) + x s}\, ds , </math>

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>\log</math> refers to the [[natural logarithm]].
In other words it is the [[Laplace transform]] of the function <math>s^s</math>.

The following real integral is equivalent to the above:

:<math>p(x) = \frac{1}{\pi} \int_0^\infty e^{-t \log(t) - x t} \sin(\pi t)\, dt.</math>

The full family of Landau distributions is obtained by extending the original distribution to a [[location-scale family]] of [[stable distributions]] with parameters <math>\alpha=1</math> and <math>\beta=1</math>,<ref>{{ cite book | last = Gentle | first = James E. | title = Random Number Generation and Monte Carlo Methods | edition = 2nd | publisher = Springer | location = New York, NY | date = 2003 | series=Statistics and Computing | isbn =978-0-387-00178-4 | doi = 10.1007/b97336 |page=196}} </ref> with [[characteristic function (probability theory)|characteristic function]]:<ref>{{cite book|last1=Zolotarev|first1=V.M.|title=One-dimensional stable distributions|date=1986|publisher=American Mathematical Society|location=Providence, R.I.|isbn=0-8218-4519-5|url-access=registration|url=https://archive.org/details/onedimensionalst00zolo_0}}</ref>

:<math>\varphi(t;\mu,c)=\exp\left(it\mu -\tfrac{2ict}{\pi}\log|t|-c|t|\right)</math>

where <math>c\in(0,\infty)</math> and <math>\mu\in(-\infty,\infty)</math>, which yields a density function:

:<math>p(x;\mu,c) = \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>

Taking <math>\mu=0</math> and <math>c=\frac{\pi}{2}</math> we get the original form of <math>p(x)</math> above.

==Properties==
[[Image:Landau_pdf.svg|300px|thumb|right|The approximation function for <math>\mu=0,\,c=1</math>]]

* Translation: If <math>X \sim \textrm{Landau}(\mu,c)\, </math> then <math> X + m \sim \textrm{Landau}(\mu + m ,c) \,</math>.
* Scaling: If <math>X \sim \textrm{Landau}(\mu,c)\, </math> then <math> aX \sim \textrm{Landau}(a\mu-\tfrac{2ac\log(a)}{\pi}, ac) \,</math>.
* Sum: If <math>X \sim \textrm{Landau}(\mu_1, c_1)</math> and <math>Y \sim \textrm{Landau}(\mu_2, c_2) \,</math> then <math> X+Y \sim \textrm{Landau}(\mu_1+\mu_2, c_1+c_2)</math>.

These properties can all be derived from the characteristic function.
Together they imply that the Landau distributions are closed under [[affine transformations]].

=== Approximations ===

In the "standard" case <math>\mu=0</math> and <math>c=\pi/2</math>, the pdf can be approximated<ref>{{Cite web|url=https://reference.wolfram.com/language/ref/LandauDistribution.html|title=LandauDistribution—Wolfram Language Documentation}}</ref> using [[Lindhard theory]] which says:

:<math>p(x+\log(x)-1+\gamma) \approx \frac{\exp(-1/x)}{x(1+x)},</math>

where <math>\gamma</math> is [[Euler's constant]].

A similar approximation <ref>{{ cite book | last1 = Behrens | first1 = S. E. | last2 = Melissinos | first2 = A.C. | title = Univ. of Rochester Preprint UR-776 (1981) }}</ref> of <math>p(x;\mu,c)</math> for <math>\mu=0</math> and <math>c=1</math> is:

:<math>p(x) \approx \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x + e^{-x}}{2}\right).</math>

==Related distributions==
* The Landau distribution is a [[stable distribution]] with stability parameter <math>\alpha</math> and skewness parameter <math>\beta</math> both equal to 1.

== References ==
{{Reflist}}

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Landau Distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Power laws]]
[[Category:Stable distributions]]
[[Category:Lev Landau]]