Editing Stable distribution (section)

==Special cases==
[[Image:Levy LdistributionPDF.png|325px|thumb|Log-log plot of symmetric centered stable distribution PDFs showing the power law behavior for large ''x''. The power law behavior is evidenced by the straight-line appearance of the PDF for large ''x'', with the slope equal to <math>-(\alpha+1)</math>. (The only exception is for <math>\alpha = 2</math>, in black, which is a normal distribution.)]]
[[Image:Levyskew LdistributionPDF.png|325px|thumb|Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large ''x''. Again the slope of the linear portions is equal to <math>-(\alpha+1)</math>]]

There is no general analytic solution for the form of ''f''(''x''). There are, however, three special cases which can be expressed in terms of [[elementary functions]] as can be seen by inspection of the [[Characteristic function (probability theory)|characteristic function]]:<ref name=":0" /><ref name=":1" /><ref>{{Cite book|title = Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance|last1 = Samorodnitsky|first1 = G.|publisher = CRC Press|year = 1994|isbn = 9780412051715|url = https://www.crcpress.com/Stable-Non-Gaussian-Random-Processes-Stochastic-Models-with-Infinite-Variance/Samoradnitsky-Taqqu/9780412051715|last2 = Taqqu|first2 = M.S.}}</ref>

* For <math>\alpha = 2</math> the distribution reduces to a [[Gaussian distribution]] with variance ''σ''<sup>2</sup> = 2''c''<sup>2</sup> and mean ''μ''; the skewness parameter <math>\beta</math> has no effect.
* For <math>\alpha = 1</math> and <math>\beta = 0</math> the distribution reduces to a [[Cauchy distribution]] with scale parameter ''c'' and shift parameter ''μ''.
* For <math>\alpha = 1/2</math> and <math>\beta = 1</math> the distribution reduces to a [[Lévy distribution]] with scale parameter ''c'' and shift parameter ''μ''.

Note that the above three distributions are also connected, in the following way: A standard Cauchy random variable can be viewed as a [[Compound probability distribution|mixture]] of Gaussian random variables (all with mean zero), with the variance being drawn from a standard Lévy distribution. And in fact this is a special case of a more general theorem (See p.&nbsp;59 of <ref name=":2">{{Cite book|url=http://eprints.nottingham.ac.uk/11194/|title=Continuous and discrete properties of stochastic processes|last= Lee|first=Wai Ha|publisher=PhD thesis, University of Nottingham|year=2010}}</ref>) which allows any symmetric alpha-stable distribution to be viewed in this way (with the alpha parameter of the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to one).

A general closed form expression for stable PDFs with rational values of <math>\alpha</math> is available in terms of [[Meijer G-function]]s.<ref>{{Cite journal|title = On Representation of Densities of Stable Laws by Special Functions|journal = Theory of Probability and Its Applications|date = 1995|issn = 0040-585X|pages = 354–362|volume = 39|issue = 2|doi = 10.1137/1139025|first = V.|last = Zolotarev}}</ref> Fox H-Functions can also be used to express the stable probability density functions. For simple rational numbers, the closed form expression is often in terms of less complicated [[special functions]]. Several closed form expressions having rather simple expressions in terms of special functions are available. In the table below, PDFs expressible by elementary functions are indicated by an ''E'' and those that are expressible by special functions are indicated by an ''s''.<ref name=":2" />

{| class="wikitable" style="text-align: center;"
|-
| ||
! colspan="7" | <math>\alpha</math>
|-
| || || 1/3 || 1/2 || 2/3 || 1   || 4/3 || 3/2 || 2
|-
! rowspan="2" | <math>\beta</math>
|  0 ||  s  ||  s  ||  s  ||   '''[[Cauchy distribution|E]]''' ||  s  ||  '''[[Holtsmark distribution|s]]'''  || rowspan="2" | '''[[Normal distribution|E]]'''
|-
|  1 ||  s  ||  '''[[Lévy distribution|E]]'''  ||  s  ||  '''[[Landau distribution|L]]'''  ||     ||  s
|-
|}

Some of the special cases are known by particular names:

* For <math>\alpha = 1</math> and <math>\beta = 1</math>, the distribution is a [[Landau distribution]] ('''L''') which has a specific usage in physics under this name.
* For <math>\alpha = 3/2</math> and <math>\beta = 0</math> the distribution reduces to a [[Holtsmark distribution]] with scale parameter ''c'' and shift parameter ''μ''.

Also, in the limit as ''c'' approaches zero or as α approaches zero the distribution will approach a [[Dirac delta function]] {{math|''δ''(''x''&nbsp;−&nbsp;''μ'')}}.