Pareto distribution of the second kind - Vose Software

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) ... Home PelicanERMSoftware WhatisPelican? Pelican-in-depthvideo WhatisEnterpriseRiskManagement? IsPelicanrightforyou? WhatmakesPelicanspecial? Videos,whitepapers,casestudies Riskmodelingsoftware ModelRisk-RiskAnalysisinExcel Tamara-ProjectRiskAnalysis Purchase Resources Frequentlyaskedquestions Exampleriskmodels Videos,whitepapers,casestudies TheVoseWikionrisk MonteCarlosimulation-asimpleexplanation Services Riskmanagementconsulting Training Technicalsupport CustomSoftwareDevelopment Company Blog Ourhistory Ourclients Contact Purchase Paretodistributionofthesecondkind Format:Pareto2(b,q)   ThisdistributionissimplyastandardPareto distribution(ofthefirstkind)butshiftedalongthex-axissothat itstartsatx=0.Thisismostreadilyapparentbystudyingthecumulative distributionfunctionsforthetwodistributions: Pareto:   Pareto2: TheonlydifferencebetweenthetwoequationsisthatxforthePareto hasbeenreplacedby(x+b)forthePareto2.Inotherwords: VosePareto2(b,q)=VosePareto(q,a) -a wherea=b,andq =q Thusbothdistributionshavethesamevarianceandshapewhena=b andq =q,butdifferentmeans. Uses APareto2(1/b,a)distribution describesthetimebetweeneventswherethenumberofrandomeventsoccurring inaunitoftimefollowsaPólya(a,b) distribution. Thisismorethananacademiccuriosity.Itiscommonlyassumedthat eventsoccurringrandomlyintimefollowaPoisson distribution,fromwhichitcanbedeterminedthatthetimebetween eventsfollowsanExponentialdistribution. However,manyeventsseemtoshowsomeclusteringintheirtiming,which meansthattherearemoreeventsclosertogetherandfurtherapartthan theExponentialdistributionwouldpredict.Fittingthetimebetweenevents usingbothaPareto2andanExponentialdistribution,thencomparingwhich distributionfitsbetter,allowsonetoassesswhetheraPólya oraPoissonshouldbeusedformodelingfrequency. Onemightargue-whynotjustfitthefrequencydatainthefirstplace? Theproblemwithfrequencydataisthatitisinformation-poor.Ifwe knowwheneventsoccurred,wecansummarizehowmanyoccurredineach timeinterval,butthereverseisnotpossible.Thus,byfittingdistributions totimebetweeneventswearemakingthemostoftheavailabledataand aremorereadilyabletodeterminewhetheraPoissonorPólya processismoreappropriate.   ModelRisk functionsaddedtoMicrosoftExcelfortheParetodistributionofthe SecondKind VosePareto2 generatesrandomvaluesfromthisdistributionforMonte Carlosimulation,orcalculatesapercentileifusedwitha U parameter. VosePareto2Object constructsadistributionobjectforthisdistribution. VosePareto2Prob returnstheprobabilitydensityorcumulativedistributionfunctionfor thisdistribution. VosePareto2Prob10 returnsthelog10oftheprobabilitydensityorcumulativedistribution function. VosePareto2Fit generatesvaluesfromthisdistributionfittedtodata,orcalculates apercentilefromthefitteddistribution. VosePareto2FitObject constructsadistributionobjectofthisdistributionfittedtodata. VosePareto2FitP returnstheparametersofthisdistributionfittedtodata.   Paretodistributionofthesecondkindequations   ©VoseSoftware™2017. 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