The Pareto Distribution was named after Italian economist and sociologist Vilfredo Pareto. · Pareto observed that 80% of the country's wealth was concentrated in ...ParetoDistributionThe"80-20Rule"Home›Resources›Knowledge›Economics›ParetoDistributionWhatisParetoDistribution?TheParetoDistributionwasnamedafterItalianeconomistandsociologistVilfredoPareto.ItissometimesreferredtoastheParetoPrincipleorthe80-20Rule.TheParetoDistributionisusedindescribingsocial,scientific,andgeophysicalphenomenainsociety.Paretocreatedamathematicalformulaintheearly20thcenturythatdescribedtheinequalitiesinwealthdistributionEconomicInequalityEconomicinequalitymostoftenreferstodisparitiesinwealthandincomethatmayexistincertainsocieties.Economicinequalityisametricthatmanyjurisdictionsandgovernmentsmonitorinordertoassesstheimpactofpolicychanges.thatexistedinhisnativecountryofItaly.  Paretoobservedthat80%ofthecountry’swealthwasconcentratedinthehandsofonly20%ofthepopulation.ThetheoryisnowappliedinmanydisciplinessuchasincomesAnnualIncomeAnnualincomeisthetotalvalueofincomeearnedduringafiscalyear.Grossannualincomereferstoallearningsbeforeanydeductionsare,productivity,populations,andothervariables.TheParetodistributionservestoshowthatthelevelofinputsandoutputsisnotalwaysequal. HistoryofParetoDistributionTheParetoDistributionprinciplewasfirstemployedinItalyintheearly20thcenturytodescribethedistributionofwealthamongthepopulation.In1906,VilfredoParetointroducedtheconceptoftheParetoDistributionwhenheobservedthat20%ofthepeapodswereresponsiblefor80%ofthepeasplantedinhisgarden.HerelatedthisphenomenontothenatureofwealthdistributioninItaly,andhefoundthat80%ofthecountry’swealthwasownedbyabout20%ofitspopulation.Intermsoflandownership,theItalianobservedthat80%ofthelandwasownedbyahandfulofwealthycitizens,whocomprisedabout20%ofthepopulation.ThedefinitionoftheParetoDistributionwaslaterexpandedinthe1940sbyDr.JosephM.Juran,aprominentproductqualityguru.JuranappliedtheParetoprincipletoqualitycontrolforbusinessproductionto

A Pareto2 (1/b, a) distribution describes the time between events where the number of random events occurring in a unit of time follows a Pólya (a,b) distribution.HomePelicanERMSoftwareWhatisPelican?Pelican-in-depthvideoWhatisEnterpriseRiskManagement?IsPelicanrightforyou?WhatmakesPelicanspecial?Videos,whitepapers,casestudiesRiskmodelingsoftwareModelRisk-RiskAnalysisinExcelTamara-ProjectRiskAnalysisPurchaseResourcesFrequentlyaskedquestionsExampleriskmodelsVideos,whitepapers,casestudiesTheVoseWikionriskMonteCarlosimulation-asimpleexplanationServicesRiskmanagementconsultingTrainingTechnicalsupportCustomSoftwareDevelopmentCompanyBlogOurhistoryOurclientsContactPurchaseParetodistributionofthesecondkindFormat:Pareto2(b,q) ThisdistributionissimplyastandardParetodistribution(ofthefirstkind)butshiftedalongthex-axissothatitstartsatx=0.Thisismostreadilyapparentbystudyingthecumulativedistributionfunctionsforthetwodistributions:Pareto: Pareto2:TheonlydifferencebetweenthetwoequationsisthatxfortheParetohasbeenreplacedby(x+b)forthePareto2.Inotherwords:VosePareto2(b,q)=VosePareto(q,a)-awherea=b,andq=qThusbothdistributionshavethesamevarianceandshapewhena=bandq=q,butdifferentmeans.UsesAPareto2(1/b,a)distributiondescribesthetimebetweeneventswherethenumberofrandomeventsoccurringinaunitoftimefollowsaPólya(a,b)distribution.Thisismorethananacademiccuriosity.ItiscommonlyassumedthateventsoccurringrandomlyintimefollowaPoissondistribution,fromwhichitcanbedeterminedthatthetimebetweeneventsfollowsanExponentialdistribution.However,manyeventsseemtoshowsomeclusteringintheirtiming,whichmeansthattherearemoreeventsclosertogetherandfurtherapartthantheExponentialdistributionwouldpredict.FittingthetimebetweeneventsusingbothaPareto2andanExponentialdistribution,thencomparingwhichdistributionfitsbetter,allowsonetoassesswhetheraPólyaoraPoissonshouldbeusedformodelingfrequency.Onemightargue-whynotjustfitthefrequencydatainthefirstplace?Theproblemwithfrequencydataisthatitisinformation-poor.Ifweknowwheneventsoccur

Pareto Distribution. DOWNLOAD Mathematica Notebook ParetoDistribution. The distribution with probability density function and distribution function ...AlgebraAppliedMathematicsCalculusandAnalysisDiscreteMathematicsFoundationsofMathematicsGeometryHistoryandTerminologyNumberTheoryProbabilityandStatisticsRecreationalMathematicsTopologyAlphabeticalIndexInteractiveEntriesRandomEntryNewinMathWorldMathWorldClassroomAboutMathWorldContributetoMathWorldSendaMessagetotheTeamMathWorldBookWolframWebResources »13,769entriesLastupdated:TueMay182021Created,developed,andnurtured by Eric WeissteinatWolfram ResearchProbabilityandStatistics > StatisticalDistributions > ContinuousDistributions >HistoryandTerminology > WolframLanguageCommands >InteractiveEntries > InteractiveDemonstrations >ParetoDistributionThedistributionwithprobabilitydensityfunctionanddistributionfunction(1)(2)definedovertheinterval.ItisimplementedintheWolframLanguageasParetoDistribution[k,alpha].Thethrawmomentis(3)for,givingthefirstfewas(4)(5)(6)(7)Thethcentralmomentis(8)(9)forandwhereisagammafunction,isaregularizedhypergeometricfunction,andisabetafunction,givingthefirstfewas(10)(11)(12)Themean,variance,skewness,andkurtosisexcessaretherefore(13)(14)(15)(16)REFERENCES:vonSeggern,D.CRCStandardCurvesandSurfaces.BocaRaton,FL:CRCPress,p. 252,1993.ReferencedonWolfram|Alpha:ParetoDistributionCITETHISAS:Weisstein,EricW."ParetoDistribution."FromMathWorld--AWolframWebResource.https://mathworld.wolfram.com/ParetoDistribution.htmlWolframWebResourcesMathematica »The#1toolforcreatingDemonstrationsandanythingtechnical.Wolfram|Alpha »Exploreanythingwiththefirstcomputationalknowledgeengine.WolframDemonstrationsProject »Explorethousandsoffreeapplicationsacrossscience,mathematics,engineering,technology,business,art,finance,socialsciences,andmore.Computerbasedmath.org »Jointheinitiativeformodernizingmatheducation.OnlineIntegralCalculator »SolveintegralswithWolfram|Alpha.Step-by-stepSolutions »Walkthroughhomeworkproblemsstep-by-stepfrom

Definition. The probability density function for the generalized Pareto distribution with shape parameter k ≠ 0, scale parameter σ, and threshold parameter θ, is.SkiptocontentDocumentationHelpCenterDocumentationSearchSupportSupportMathWorksSearchMathWorks.comMathWorksSupportCloseMobileSearchOpenMobileSearchOff-CanvasNavigationMenuToggleDocumentationHomeStatisticsandMachineLearningToolboxProbabilityDistributionsContinuousDistributionsGeneralizedParetoDistributionGeneralizedParetoDistributionOnthispageDefinitionBackgroundParametersExamplesComputeGeneralizedParetoDistributionpdfReferencesSeeAlsoRelatedTopicsDocumentationAllExamplesFunctionsBlocksAppsVideosAnswersTrialSoftwareTrialSoftwareProductUpdatesProductUpdatesResourcesDocumentationAllExamplesFunctionsBlocksAppsVideosAnswersMainContentGeneralizedParetoDistributionDefinitionTheprobabilitydensityfunctionforthegeneralizedParetodistributionwithshapeparameterk≠0,scaleparameterσ,andthresholdparameterθ,isy​​  =​ f(x|k,σ,θ)=​​​​​​ (1σ)(1+k(x−θ)σ)−1−1kforθ0,orforθ0andθ=σ/k,thegeneralizedParetodistributionisequivalenttotheParetodistributionwithascaleparameterequaltoσ/kandashapeparameterequalto1/k.BackgroundLiketheexponentialdistribution,thegeneralizedParetodistributionisoftenusedtomodelthetailsofanotherdistribution.Forexample,youmighthavewashersfromamanufacturingprocess.Ifrandominfluencesintheprocessleadtodifferencesinthesizesofthewashers,astandardprobabilitydistribution,suchasthenormal,couldbeusedtomodelthosesizes.However,whilethenormaldistributionmightbeagoodmodelnearitsmode,itmightnotbeagoodfittorealdatainthetailsandamorecomplexmodelmightbeneededtodescribethefullrangeofthedata.Ontheotherhand,onlyrecordingthesizesofwasherslarger(orsmaller)thanacertainthresholdmeansyoucanfitaseparatemodeltothosetaildata,whichareknownasexceedances.Youcanusethegener

The Pareto distribution has a very long right-hand tail. It is often applied in the study of socioeconomic data, including the distribution of income, firm size, ...HomeParetoDistributionTheParetodistributionisapower-lawprobabilitydistributionthatisusedindescriptionofsocial,scientific,geophysical,actuarial,andmanyothertypesofobservablephenomena.Originallyappliedtodescribingthedistributionofwealthinasociety,fittingthetrendthatalargeportionofwealthisheldbyasmallfractionofthepopulation,theParetodistributionhascolloquiallybecomeknownandreferredtoastheParetoprinciple,or“80-20rule”.Thisrulestatesthat,forexample,80%ofthewealthofasocietyisheldby20%ofitspopulation.However,theParetodistributiononlyproducesthisresultforaparticularpowervalue,\(\alpha\)(\(\alpha\)=log45≈1.16).While\(\alpha\)isvariable,empiricalobservationhasfoundthe80-20distributiontofitawiderangeofcases,includingnaturalphenomenaandhumanactivities(Wikipedia).Theprobabilitydensityfunctionofthecommonparetodistibutionis:\(f(x)=(x_{m}/x)^\alpha\)where\(x_{m}\)isthe(necessarilypositive)minimumpossiblevalueofX.Itisscaleparameter,andαisapositiveparameter.Itisashapeparameter.x=seq(1,5,0.05)a3=3/x^(4)a2=2/x^(3)a1=1/x^(2)df=bind_rows(data_frame(x=x,p=a3,alpha='3'),data_frame(x=x,p=a2,alpha='2'),data_frame(x=x,p=a1,alpha='1'))df%>%ggplot(aes(x=x,y=p,group=alpha))+geom_line(aes(color=alpha),size=1)+xlim(1,5)+ggtitle('Fig.1.ParetoProbabilitydensityfunctionsforvariousshapeparameters')+ylab('pr(X=x)')+theme(axis.title=element_text(size=14),plot.title=element_text(size=16,colour="purple"),axis.text=element_text(size=14))TheParetodistributionhasaverylongright-handtail.Itisoftenappliedinthestudyofsocioeconomicdata,includingthedistributionofincome,firmsize,population,andstockpricefluctuations.TheParetodistributiontakesvaluesonthepositiverealline.Allvaluesmustbelargerthanthe“location”parameterη,whichisreallyathresholdparameter.TherearethreekindsofParetodistributions.TheonedescribedhereistheParetodistributionofthefirstkind.ThePareto

In Statistical theory, inclusion of an additional parameter to standard distributions is a usual practice. In this study, a new distribution referred to ...BrowseSubjectAreas?ClickthroughthePLOStaxonomytofindarticlesinyourfield.FormoreinformationaboutPLOSSubjectAreas,clickhere.ArticleAuthorsMetricsCommentsMediaCoverageReaderComments(0)FiguresFiguresAbstractInStatisticaltheory,inclusionofanadditionalparametertostandarddistributionsisausualpractice.Inthisstudy,anewdistributionreferredtoasAlpha-PowerParetodistributionisintroducedbyincludinganextraparameter.Severalpropertiesoftheproposeddistribution,includingmomentgeneratingfunction,mode,quantiles,entropies,meanresiduallifefunction,stochasticordersandorderstatisticsareobtained.Parametersoftheproposeddistributionhavebeenestimatedusingmaximumlikelihoodestimationtechnique.Tworealdatasetshavebeenconsideredtoexaminetheusefulnessoftheproposeddistribution.IthasbeenobservedthattheproposeddistributionoutperformsdifferentvariantsofParetodistributiononthebasisofmodelselectioncriteria.Citation:IhtishamS,KhalilA,ManzoorS,KhanSA,AliA(2019)Alpha-PowerParetodistribution:Itspropertiesandapplications.PLoSONE14(6):e0218027.https://doi.org/10.1371/journal.pone.0218027Editor:HaroldoV.Ribeiro,UniversidadeEstadualdeMaringa,BRAZILReceived:March1,2019;Accepted:May23,2019;Published:June12,2019Copyright:©2019Ihtishametal.ThisisanopenaccessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited.DataAvailability:AllrelevantdataarewithinthepaperanditsSupportingInformationfiles.Funding:Theauthorsreceivednospecificfundingforthiswork.Competinginterests:Theauthorshavedeclaredthatnocompetinginterestsexist.IntroductionForthelastfewdecades,improvementoverstandarddistributionshasbecomeacommonpracticeinstatisticaltheory.Usually,anadditionalparameterisaddedbyusinggeneratorsorexistingdistributionsarecombinedtoobtainnewdistributions[1]

柏拉圖分布[編輯] ... 柏拉圖分布(Pareto distribution)是以義大利經濟學家維爾弗雷多·柏拉圖命名的。

是從大量真實世界的現象中發現的冪定律分布。

這個分布在經濟學 ...柏拉圖分布維基百科，自由的百科全書跳至導覽跳至搜尋此條目需要擴充。

(2013年11月8日)請協助改善這篇條目，更進一步的訊息可能會在討論頁或擴充請求中找到。

請在擴充條目後將此模板移除。

帕累托分布機率密度函數累積分布函數母數xm>0k>0值域x∈[xm;+∞){\displaystylex\in[x_{\mathrm{m}};+\infty)\!}機率密度函數kxmkxk+1{\displaystyle{\frac{k\,x_{\mathrm{m}}^{k}}{x^{k+1}}}\!}累積分布函數1−(xmx)k{\displaystyle1-\left({\frac{x_{\mathrm{m}}}{x}}\right)^{k}\!}期望值kxmk−1{\displaystyle{\frac{k\,x_{\mathrm{m}}}{k-1}}\!}，k>1{\displaystylek>1}中位數xm2k{\displaystylex_{\mathrm{m}}{\sqrt[{k}]{2}}}眾數xm{\displaystylex_{\mathrm{m}}\,}變異數xm2k(k−1)2(k−2){\displaystyle{\frac{x_{\mathrm{m}}^{2}k}{(k-1)^{2}(k-2)}}\!}，k>2{\displaystylek>2}偏度2(1+k)k−3k−2k{\displaystyle{\frac{2(1+k)}{k-3}}\,{\sqrt{\frac{k-2}{k}}}\!}，k>3{\displaystylek>3}峰度6(k3+k2−6k−2)k(k−3)(k−4){\displaystyle{\frac{6(k^{3}+k^{2}-6k-2)}{k(k-3)(k-4)}}\!}，k>4{\displaystylek>4}熵ln⁡(kxm)−1k−1{\displaystyle\ln\left({\frac{k}{x_{\mathrm{m}}}}\right)-{\frac{1}{k}}-1\!}動差母函數未定義特徵函數k(−ixmt)kΓ(−k,−ixmt){\displaystylek(-ix_{\mathrm{m}}t)^{k}\Gamma(-k,-ix_{\mathrm{m}}t)\,}柏拉圖分布(Paretodistribution)是以義大利經濟學家維爾弗雷多·柏拉圖命名的。

是從大量真實世界的現象中發現的冪定律分布。

這個分布在經濟學以外，也被稱為布拉德福分布。

在柏拉圖分布中，如果X是一個隨機變數，則X的機率分布如下面的公式所示：P(X>x)=(xxmin)−k{\displaystyle{\rm{P}}(X>x)=\left({\frac{x}{x_{\min}}}\right)^{-k}}其中x是任何一個大於xmin的數，xmin是X最小的可能值（正數），k是為正的母數。

柏拉圖分布曲線族是由兩個數量母數化的：xmin和k。

分布密度則為p(x)={0,if xxmin.{\displaystylep(x)=\left\{{\begin{matrix}0,&{\mbox{if}}xx_{\min}.\end{matrix}}\right.}柏拉圖分布屬於連續機率分布。

「齊夫定律」,也稱為「zeta分布」,也可以被認為是在離散機率分布中的柏拉圖分布。

一個遵守柏拉圖分布的隨機變數的期望值為xminkk−1{\displaystylex_{\min}\;k\overk-1}(如果k≤1{\displaystylek\leq1},期望值為無窮大)且隨機變數的標準差為xmink−1kk−2{\displaystyle{x_{\min}\overk-1}{\sqrt{k\overk-2}}}(如果k≤2{\displaystylek\leq2},標準差不存在)。

被認為大致是柏拉圖分布的例子有：財富在個人之間的分布人類居住區的大小對維基百科條目的訪問接近絕對零度時，玻色–愛因斯坦凝聚的團簇在網際網路流量中文件尺寸的分布油田的石油儲備數量龍捲風帶來的災

The Pareto Principle is derived from the Pareto distribution and is used to illustrate that many things are not distributed evenly. Originally written to ...ShareonProbabilityDistributions>WhatistheParetoDistribution?TheParetodistributionisaskeweddistributionwithheavy,or“slowlydecaying”tails(i.e.muchofthedataisinthetails).TheParetodistribution.Image:Danvildanvil|wikimediacommonsTheParetodistribution(createdbythe19thCenturyItalianeconomistVilfredoPareto)isdefinedbyashapeparameter,α(alsocalledaslopeparameterorParetoIndex)andalocationparameter,X.Ithastwomainapplications:Tomodelthedistributionofincomes.Tomodelthedistributionofcitypopulations.However,itcanbeusedinavarietyofothersituations.Forexample,itcanbeusedtomodelthelifetimeofamanufactureditemwithacertainwarrantyperiod.TheParetodistributionisexpressedas:F(x)=1–(k/x)αwherexistherandomvariablekisthelowerboundofthedataαistheshapeparameterYoumightalsoseethiswrittenas:F(x)=1–(kλk/xk+1)αWhenusedtomodelincomedistribution,thisparticularversionoftheformulahasλastheminimumincomeandkasthedistributionofincome.TheSurvivalFunctionMosttextsontheParetofunctionmentiona“survivalfunction,”althoughthisissometimesalsocalledatailfunctionorreliabilityfunction.ThisisjusttheprobabilityofvaluesgreaterthanX.Forexample,youmaybelookingathouseholdincomeintheUnitedStatesandwanttoknowwhatproportionofhouseholdincomeisgreaterthan$1,000,000.TheParetoPrincipleTheParetoPrincipleisderivedfromtheParetodistributionandisusedtoillustratethatmanythingsarenotdistributedevenly.Originallywrittentostatethat20%ofthepopulationholds80%ofthewealth,itcanbeappliedmoreuniversally.Forexample,1%ofthepopulationholds99%ofthewealth.However,itcanbeusedtomodelanygeneralsituationwheresituationsarenotevenlydistributed.Forexample,thetop20%ofworkersmightproduce80%ofoutput.Next:TheParetoPrinciple.CITETHISAS:StephanieGlen."ParetoDistributionDefinition"FromStatisticsHowTo.com:ElementaryStatisticsfortherestofus!https://www.statisticshowto.com/pareto-distribution/----------------------

The Pareto Type I distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. When this distribution is used to ...ParetodistributionFromWikipedia,thefreeencyclopediaJumptonavigationJumptosearchProbabilitydistributionParetoTypeIProbabilitydensityfunctionParetoTypeIprobabilitydensityfunctionsforvariousα{\displaystyle\alpha}withxm=1.{\displaystylex_{\mathrm{m}}=1.}Asα→∞,{\displaystyle\alpha\rightarrow\infty,}thedistributionapproachesδ(x−xm),{\displaystyle\delta(x-x_{\mathrm{m}}),}whereδ{\displaystyle\delta}istheDiracdeltafunction.CumulativedistributionfunctionParetoTypeIcumulativedistributionfunctionsforvariousα{\displaystyle\alpha}withxm=1.{\displaystylex_{\mathrm{m}}=1.}Parametersxm>0{\displaystylex_{\mathrm{m}}>0}scale(real)α>0{\displaystyle\alpha>0}shape(real)Supportx∈[xm,∞){\displaystylex\in[x_{\mathrm{m}},\infty)}PDFαxmαxα+1{\displaystyle{\frac{\alphax_{\mathrm{m}}^{\alpha}}{x^{\alpha+1}}}}CDF1−(xmx)α{\displaystyle1-\left({\frac{x_{\mathrm{m}}}{x}}\right)^{\alpha}}Mean{∞for α≤1αxmα−1for α>1{\displaystyle{\begin{cases}\infty&{\text{for}}\alpha\leq1\\{\dfrac{\alphax_{\mathrm{m}}}{\alpha-1}}&{\text{for}}\alpha>1\end{cases}}}Medianxm2α{\displaystylex_{\mathrm{m}}{\sqrt[{\alpha}]{2}}}Modexm{\displaystylex_{\mathrm{m}}}Variance{∞for α≤2xm2α(α−1)2(α−2)for α>2{\displaystyle{\begin{cases}\infty&{\text{for}}\alpha\leq2\\{\dfrac{x_{\mathrm{m}}^{2}\alpha}{(\alpha-1)^{2}(\alpha-2)}}&{\text{for}}\alpha>2\end{cases}}}Skewness2(1+α)α−3α−2α for α>3{\displaystyle{\frac{2(1+\alpha)}{\alpha-3}}{\sqrt{\frac{\alpha-2}{\alpha}}}{\text{for}}\alpha>3}Ex.kurtosis6(α3+α2−6α−2)α(α−3)(α−4) for α>4{\displaystyle{\frac{6(\alpha^{3}+\alpha^{2}-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}}{\text{for}}\alpha>4}Entropylog⁡((xmα)e1+1α){\displaystyle\log\left(\left({\frac{x_{\mathrm{m}}}{\alpha}}\right)\,e^{1+{\tfrac{1}{\alpha}}}\right)}MGFdoesnotexistCFα(−ixmt)αΓ(−α,−ixmt){\displaystyle\alpha(-ix_{\mathrm{m}}t)^{\alpha}\Gamma(-\alpha,-ix_{\mathrm{m}}