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| {{Infobox probability distribution
| | I'm a 33 years old and study at the college (Comparative Politics).<br>In my spare time I learn Russian. I have been twicethere and look forward to returning anytime soon. I like to read, preferably on my kindle. I really love to watch Sons of Anarchy and Doctor Who as well as documentaries about anything geological. I like Reading.<br><br>Check out my web-site - [http://www.sexytgps.net turk porno |] |
| | name = Delaporte
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| | type = discrete
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| | pdf_image = [[File:DelaportePMF.svg|325px|Plot of the PMF for various Delaporte distributions.]]<br /> When <math>\alpha</math> and <math>\beta</math> are 0, the distribution is the Poisson.<br />When <math>\lambda</math> is 0, the distribution is the negative binomial.
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| | cdf_image = [[File:DelaporteCDF.svg|325px|Plot of the PMF for various Delaporte distributions.]]<br /> When <math>\alpha</math> and <math>\beta</math> are 0, the distribution is the Poisson.<br />When <math>\lambda</math> is 0, the distribution is the negative binomial.
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| | notation =
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| | parameters = <math>\lambda > 0</math> (fixed mean)
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| <math>\alpha, \beta > 0</math> (parameters of variable mean)
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| | support = <math>k \in \{0, 1, 2, \ldots\}</math>
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| | pdf = <math>\sum_{i=0}^k\frac{\Gamma(\alpha + i)\beta^i\lambda^{k-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(k-i)!}</math>
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| | cdf = <math>\sum_{j=0}^k\sum_{i=0}^j\frac{\Gamma(\alpha + i)\beta^i\lambda^{j-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(j-i)!}</math>
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| | mean = <math>\lambda + \alpha\beta</math>
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| | median =
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| | mode = <math>\begin{cases}z, z+1 & \{z \in \mathbb{Z}\}:\; z = (\alpha-1)\beta+\lambda\\ \lfloor z \rfloor & \textrm{otherwise}\end{cases}</math>
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| | variance = <math>\lambda + \alpha\beta(1+\beta)</math>
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| | skewness = See [[#Properties]]
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| | kurtosis = See [[#Properties]]
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| | entropy =
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| | mgf =
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| | cf =
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| | pgf =
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| | fisher =
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| }}
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| The '''Delaporte distribution''' is a [[discrete probability distribution]] that has received attention in [[actuarial science]].<ref name = "EAS">{{cite encyclopedia
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| | last = Panjer
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| | first = Harry H.
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| | editor1-last = Teugels
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| | editor1-first = Jozef L.
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| | editor2-first = Bjørn
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| | editor2-last = Sundt
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| | encyclopedia = Encyclopedia of Actuarial Science
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| | title = Discrete Parametric Distributions
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| | year = 2006
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| | publisher = [[John Wiley & Sons]]
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| | isbn = 978-0-470-01250-5
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| | doi = 10.1002/9780470012505.tad027
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| }}
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| </ref><ref name = "UDD"/> It can be defined using the [[convolution]] of a [[negative binomial distribution]] with a [[Poisson distribution]].<ref name = "UDD">{{cite book
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| | last1 = Johnson
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| | first1 = Norman Lloyd
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| | author1-link = Norman Lloyd Johnson
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| | last2 = Kemp
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| | first2 = Adrienne W.
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| | last3 = Kotz
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| | first3 = Samuel
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| | author3-link = Samuel Kotz
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| | title = Univariate discrete distributions
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| | edition = Third
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| | year = 2005
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| | publisher = [[John Wiley & Sons]]
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| | isbn = 978-0-471-27246-5
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| | pages = 241–242
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| }}
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| </ref> Just as the [[negative binomial distribution]] can be viewed as a Poisson distribution where the mean parameter is itself a random variable with a [[gamma distribution]], the Delaporte distribution can be viewed as a [[compound distribution]] based on a Poisson distribution, where there are two components to the mean parameter: a fixed component, which has the <math>\lambda</math> parameter, and a gamma-distributed variable component, which has the <math>\alpha</math> and <math>\beta</math> parameters.<ref name = "Vose">{{cite book
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| | last1 = Vose
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| | first1 = David
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| | title = Risk analysis: a quantitative guide
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| | edition = Third, illustrated
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| | year = 2008
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| | publisher = [[John Wiley & Sons]]
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| | isbn = 978-0-470-51284-5
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| | lccn = 2007041696
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| | pages = 618–619
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| }}
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| </ref> The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959,<ref name = "DP">{{cite journal
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| | last1 = Delaporte
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| | first1 = Pierre J.
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| | year = 1960
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| | month =
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| | title = Quelques problèmes de statistiques mathématiques poses par l’Assurance Automobile et le Bonus pour non sinistre
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| | trans_title = Some problems of mathematical statistics as related to automobile insurance and no-claims bonus
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| | journal = Bulletin Trimestriel de l'Institut des Actuaires Français
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| | volume = 227
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| | pages = 87–102
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| | language = French
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| }}
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| </ref> although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders,<ref name = "Luders">{{cite journal
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| | last1 = von Lüders
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| | first1 = Rolf
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| | year = 1934
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| | title = Die Statistik der seltenen Ereignisse
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| | trans_title = The statistics of rare events
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| | journal = [[Biometrika]]
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| | volume = 26
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| | pages = 108–128
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| | language = German
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| | doi=10.1093/biomet/26.1-2.108
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| | jstor=2332055
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| }}
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| </ref> where it was called the Formel II distribution.<ref name = "UDD" /> | |
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| ==Properties==
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| The [[skewness]] of the Delaporte distribution is:
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| <math>
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| \frac{\lambda + \alpha\beta(1+3\beta+2\beta^2)}{\left(\lambda + \alpha\beta(1+\beta)\right)^{\frac{3}{2}}}
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| </math>
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| The [[kurtosis|excess kurtosis]] of the distribution is:
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| <math>
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| \frac{\lambda+3\lambda^2+\alpha\beta(1+6\lambda+6\lambda\beta+7\beta+12\beta^2+6\beta^3+3\alpha\beta+6\alpha\beta^2+3\alpha\beta^3)}{\left(\lambda + \alpha\beta(1+\beta)\right)^2}
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| </math>
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{cite journal|
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| last1=Murat |first1= M.
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| |last2=Szynal |first2= D.
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| |title= On moments of counting distributions satisfying the k'th-order recursion and their compound distributions
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| |journal=Journal of Mathematical Sciences
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| |year=1998
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| |pages=4038–4043
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| |volume= 92 |issue= 4
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| |doi= 10.1007/BF02432340 }}
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| ==External links==
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| *[http://vosesoftware.com/ModelRiskHelp/index.htm#Distributions/Discrete_distributions/Delaporte_distribution.htm Delaporte distribution] at Vose Software. Details of derivation.
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| {{ProbDistributions|discrete-infinite}}
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| {{Common univariate probability distributions|state=collapsed}}
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| {{DEFAULTSORT:Delaporte distribution}}
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| [[Category:Discrete distributions]]
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| [[Category:Compound distributions]]
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| [[Category:Probability distributions]]
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I'm a 33 years old and study at the college (Comparative Politics).
In my spare time I learn Russian. I have been twicethere and look forward to returning anytime soon. I like to read, preferably on my kindle. I really love to watch Sons of Anarchy and Doctor Who as well as documentaries about anything geological. I like Reading.
Check out my web-site - turk porno |