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{{Probability distribution
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
| name      =Lomax
| type      =density
| pdf_image  =
| cdf_image  =
| parameters =
<math>\lambda >0 </math> [[scale parameter|scale]] (real)<br />
<math>\alpha > 0 </math> [[shape parameter|shape]] (real)
| support    =<math> x \ge 0 </math>
| pdf        =<math> {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}</math>
| cdf        =<math> 1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}</math>
| mean      =<math> {\lambda \over {\alpha -1}} \text{ for } \alpha > 1</math><br /> Otherwise undefined
| median    =<math>\lambda (\sqrt[\alpha]{2} - 1)</math>
| mode       = 0
| variance  =<math> {{\lambda^2 \alpha} \over {(\alpha-1)^2(\alpha-2)}} \text{ for } \alpha > 2 </math><br /><math> \infty \text{ for } 1 < \alpha \le 2 </math> <br /> Otherwise undefined
| skewness  =<math>\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3\,</math>
| kurtosis  =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4\,</math>
| entropy    =
| mgf        =
| char      =
}}


The '''Lomax distribution''', conditionally also called the '''[[Pareto_distribution#Pareto types I–IV|Pareto Type II distribution]]''', is a [[heavy tail|heavy-tail]] [[probability distribution]] often used in business, economics, and actuarial modeling.<ref>Lomax, K. S. (1954) "Business Failures; Another example of the analysis of failure data". ''[[Journal of the American Statistical Association]]'', 49, 847–852. {{jstor|2281544}}</ref><ref>Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) ''Continuous Univariate Distributions, Volume 1'', 2nd Edition, Wiley. ISBN 0-471-58495-9 (pages 575, 602)</ref>  It is named after K.&nbsp;S.&nbsp;Lomax. It is essentially a [[Pareto distribution]] that has been shifted so that its support begins at zero.<ref>Van Hauwermeiren M and Vose D (2009). [http://www.vosesoftware.com/content/ebook.pdf '' A Compendium of Distributions''] [ebook]. Vose Software, Ghent, Belgium. Available at www.vosesoftware.com. Accessed 07/07/11</ref>
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


== Characterization ==
Registered users will be able to choose between the following three rendering modes:  
=== Probability density function ===
The [[probability density function]] (pdf) for the Lomax distribution is given by
:<math> p(x) = {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}, \qquad x \geq 0,
</math>
with shape parameter <math>\alpha>0</math> and scale parameter <math>\lambda>0</math>. The density can be rewritten in such a way that more clearly shows the relation to the [[Pareto distribution|Pareto Type I distribution]]. That is:
:<math> p(x) = {{\alpha \lambda^\alpha} \over { (x+\lambda)^{\alpha+1}}}</math>.


===Differential equation===
'''MathML'''
The pdf of the Lomax distribution is a solution to the following [[differential equation]]:
:<math forcemathmode="mathml">E=mc^2</math>
:<math>\left\{\begin{array}{l}
(\gamma +x) p'(x)+(\alpha +1) p(x)=0, \\
p(0)=\frac{\alpha}{\gamma}
\end{array}\right\}
</math>


== Relation to the Pareto distribution ==
<!--'''PNG'''  (currently default in production)
The Lomax distribution is a [[Pareto distribution|Pareto Type I distribution]] shifted so that its support begins at zero. Specifically:
:<math forcemathmode="png">E=mc^2</math>
:<math>\text{If } Y \sim \mbox{Pareto}(x_m = \lambda, \alpha), \text{ then } Y - x_m \sim \mbox{Lomax}(\lambda,\alpha).</math>


The Lomax distribution is a [[Pareto_distribution#Pareto types I–IV|Pareto Type II distribution]] with ''x''<sub>m</sub>=λ and μ=0:{{cn|date=October 2012}}
'''source'''
:<math>
:<math forcemathmode="source">E=mc^2</math> -->
\text{If } X \sim \mbox{Lomax}(\lambda,\alpha) \text{ then } X \sim \text{P(II)}(x_m = \lambda, \alpha, \mu=0).</math>


== Relation to generalized Pareto distribution ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
The Lomax distribution is a special case of the [[generalized Pareto distribution]]. Specifically:


:<math> \mu = 0,~ \xi = {1 \over \alpha},~ \sigma = {\lambda \over \alpha} .</math>
==Demos==


== Relation to q-exponential distribution ==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
The Lomax distribution is a special case of the [[q-exponential distribution]]. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by:


:<math> \alpha = { {2-q} \over {q-1}}, ~ \lambda = {1 \over \lambda_q (q-1)} .</math>


== Non-central moments ==
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


The <math>\nu</math>th non-central moment <math>E[X^\nu]</math> exists only if the shape parameter <math>\alpha</math> strictly exceeds <math>\nu</math>, when the moment has the value
==Test pages ==
:<math> E(X^\nu) = \frac{ \lambda^\nu \Gamma(\alpha-\nu)\Gamma(1+\nu)}{\Gamma(\alpha)}</math>


== See also ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Power law]]
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==References==
*[[Inputtypes|Inputtypes (private Wikis only)]]
<references />
*[[Url2Image|Url2Image (private Wikis only)]]
{{ProbDistributions|continuous-semi-infinite}}
==Bug reporting==
[[Category:Continuous distributions]]
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
[[Category:Probability distributions with non-finite variance]]
[[Category:Probability distributions]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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