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| {{Probability distribution
| | Hello from Netherlands. I'm glad to came across you. My first name is Dominique. <br>I live in a city called Woudenberg in east Netherlands.<br>I was also born in Woudenberg 24 years ago. Married in May 2006. I'm working at the backery.<br><br>My website; [http://tinyurl.com/kew25nq cheap ugg boots sale] |
| | name = Holtsmark
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| | type = continuous
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| | pdf_image = [[File:Levy distributionPDF.png|325px|Symmetric stable distributions]]<br /><small>Symmetric ''α''-stable distributions with unit scale factor; ''α''=1.5 (blue line) represents the Holtsmark distribution</small>
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| | cdf_image = [[File:Levy distributionCDF.png|325px|CDF's for symmetric ''α''-stable distributions; ''α''=3/2 represents the Holtsmark distribution]]
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| | parameters = ''c'' ∈ (0, ∞) — [[scale parameter]] <br>
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| ''μ'' ∈ (−∞, ∞) — [[location parameter]]
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| | support = ''x'' ∈ '''R'''
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| | pdf = expressible in terms of [[hypergeometric function]]s; see text
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| | cdf =
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| | mean = ''μ''
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| | median = ''μ''
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| | mode = ''μ''
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| | variance = infinite
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| | skewness = undefined
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| | kurtosis = undefined
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| | entropy =
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| | mgf = undefined
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| | char = <math>\exp\left[~it\mu\!-\!|c t|^{3/2}~\right]</math>
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| }}
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| The (one-dimensional) '''Holtsmark distribution''' is a [[continuous probability distribution]]. The Holtsmark distribution is a special case of a [[stable distribution]] with the index of stability or shape parameter <math>\alpha</math> equal to 3/2 and skewness parameter <math>\beta</math> of zero. Since <math>\beta</math> equals zero, the distribution is symmetric, and thus an example of a symmetric alpha-stable distribution. The Holtsmark distribution is one of the few examples of a stable distribution for which a closed form expression of the [[probability density function]] is known. However, its probability density function is not expressible in terms of [[elementary functions]]; rather, the probability density function is expressed in terms of [[hypergeometric functions]].
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| The Holtsmark distribution has applications in plasma physics and astrophysics.<ref name=lee/> In 1919, Norwegian physicist [[Johan Peter Holtsmark|J. Holtsmark]] proposed the distribution as a model for the fluctuating fields in plasma due to [[chaotic]] motion of charged particles.<ref>{{Cite journal
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| | doi = 10.1002/andp.19193630702
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| | volume = 363
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| | issue = 7
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| | pages = 577–630
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| | last = Holtsmark
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| | first = J.
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| | title = Uber die Verbreiterung von Spektrallinien
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| | journal = Annalen der Physik
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| | accessdate = 2009-01-06
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| | year = 1919
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| |bibcode = 1919AnP...363..577H }}</ref> It is also applicable to other types of Coulomb forces, in particular to modeling of gravitating bodies, and thus is important in astrophysics.<ref>{{Cite journal
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| | doi = 10.1086/144420
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| | issn = 0004-637X
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| | volume = 95
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| | pages = 489
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| | last = Chandrasekhar
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| | first = S.
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| | coauthors = J. von Neumann
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| | title = The Statistics of the Gravitational Field Arising from a Random Distribution of Stars. I. The Speed of Fluctuations
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| | journal = The Astrophysical Journal
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| | accessdate = 2011-03-01
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| | year = 1942
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| | url = http://adsabs.harvard.edu/doi/10.1086/144420
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| | bibcode=1942ApJ....95..489C
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| }}</ref><ref>{{Cite journal
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| | doi = 10.1103/RevModPhys.15.1
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| | volume = 15
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| | issue = 1
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| | pages = 1
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| | last = Chandrasekhar
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| | first = S.
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| | title = Stochastic Problems in Physics and Astronomy
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| | journal = Reviews of Modern Physics
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| | accessdate = 2011-03-01
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| | date = 1943-01-01
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| | url = http://link.aps.org/doi/10.1103/RevModPhys.15.1
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| | bibcode=1943RvMP...15....1C
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| }}</ref>
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| ==Characteristic function==
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| The [[characteristic function (probability theory)|characteristic function]] of a symmetric stable distribution is:
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| : <math>
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| \varphi(t;\mu,c) =
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| \exp\left[~it\mu\!-\!|c t|^\alpha~\right],
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| </math>
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| where <math>\alpha</math> is the shape parameter, or index of stability, <math>\mu</math> is the [[location parameter]], and ''c'' is the [[scale parameter]].
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| Since the Holtsmark distribution has <math>\alpha=3/2,</math> its characteristic function is:<ref name=zolotarev>{{cite book|title=One-Dimensional Stable Distributions|author=Zolotarev, V. M.|pages=1, 41|year=1986|location=Providence, RI|publisher=[[American Mathematical Society]]|isbn=978-0-8218-4519-6|url=http://books.google.com/?id=ydwt9SotnN0C&pg=PR7&dq=Vladimir+Zolotarev+One-dimensional+stable+laws#v=onepage&q=holtsmark&f=false}}</ref>
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| : <math>
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| \varphi(t;\mu,c) =
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| \exp\left[~it\mu\!-\!|c t|^{3/2}~\right] .
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| </math>
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| Since the Holtsmark distribution is a stable distribution with {{nowrap|''α'' > 1}}, <math>\mu</math> represents the [[mean]] of the distribution.<ref name=nolan>{{cite book|chapter=Basic Properties of Univariate Stable Distributions|title=Stable Distributions: Models for Heavy Tailed Data|author=Nolan, J. P.|pages=3, 15–16|url=http://lpmt-theory.wdfiles.com/local--files/michael-blog/stablePDF.pdf|year=2008|accessdate=2011-02-06}}</ref><ref>{{cite book|title=Handbook of Heavy Tailed Distributions in Finance|editor=Rachev, S. T.|chapter=Modeling Financial Data|pages=111–112|author=Nolan, J. P.|year=2003|location=Amsterdam|publisher=[[Elsevier]]|isbn=978-0-444-50896-6}}</ref> Since {{nowrap|''β'' {{=}} 0}}, <math>\mu</math> also represents the [[median]] and [[Mode (statistics)|mode]] of the distribution. And since {{nowrap|''α'' < 2}}, the [[variance]] of the Holtsmark distribution is infinite.<ref name=nolan/> All higher [[Moment (mathematics)|moments]] of the distribution are also infinite.<ref name=nolan/> Like other stable distributions (other than the normal distribution), since the variance is infinite the dispersion in the distribution is reflected by the [[scale parameter]], c. An alternate approach to describing the dispersion of the distribution is through fractional moments.<ref name=nolan/>
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| ==Probability density function==
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| In general, the [[probability density function]], ''f''(''x''), of a continuous probability distribution can be derived from its characteristic function by:
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| :<math>
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| f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty \varphi(t)e^{-ixt}\,dt .
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| </math>
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| Most stable distributions do not have a known closed form expression for their probability density functions. Only the [[normal distribution|normal]], [[Cauchy distribution|Cauchy]] and [[Lévy distribution]]s have known closed form expressions in terms of [[elementary functions]].<ref name=lee/> The Holtsmark distribution is one of two symmetric stable distributions to have a known closed form expression in terms of [[generalized hypergeometric function|hypergeometric functions]].<ref name=lee/> When <math>\mu</math> is equal to 0 and the scale parameter is equal to 1, the Holtsmark distribution has the probability density function:
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| :<math>
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| \begin{align}
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| f(x; 0, 1)
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| &= { 1 \over \pi } {\Gamma(5/3)} {}_2F_3(5/12,11/12;1/3,1/2,5/6;-4x^6/729) \\
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| & {} \quad{} - { x^2 \over 3\pi } {}_3F_4(3/4,1,5/4;2/3,5/6,7/6,4/3;-4x^6/729) \\
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| & {} \quad{} + { 7x^4 \over 81\pi } {\Gamma(4/3)} {}_2F_3(13/12,19/12;7/6,3/2,5/3;-4x^6/729) ,
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| \end{align}
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| </math>
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| where <math>{\Gamma(x)}</math> is the [[gamma function]] and <math> \;_mF_n()</math> is a [[generalized hypergeometric function|hypergeometric function]].<ref name=lee>{{cite book|title=Continuous and Discrete Properties of Stochastic Processes|author=Lee, W. H.|pages=37–39|type=PhD thesis|year=2010|location=[[University of Nottingham]]|url=http://etheses.nottingham.ac.uk/1194/1/Thesis_Wai_Ha_Lee.pdf}}</ref>
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| ==References==
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| {{reflist}}
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| *{{cite doi|10.1016/0022-4073(86)90011-7}}
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| {{ProbDistributions|continuous-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions with non-finite variance]]
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| [[Category:Power laws]]
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| [[Category:Stable distributions]]
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| [[Category:Probability distributions]]
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Hello from Netherlands. I'm glad to came across you. My first name is Dominique.
I live in a city called Woudenberg in east Netherlands.
I was also born in Woudenberg 24 years ago. Married in May 2006. I'm working at the backery.
My website; cheap ugg boots sale