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