Pareto distribution - Wikipedia

Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the ... Paretodistribution FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Probabilitydistribution ParetoTypeI ProbabilitydensityfunctionParetoTypeIprobabilitydensityfunctionsforvarious α {\displaystyle\alpha} with x m = 1. {\displaystylex_{\mathrm{m}}=1.} As α → ∞ , {\displaystyle\alpha\rightarrow\infty,} thedistributionapproaches δ ( x − x m ) , {\displaystyle\delta(x-x_{\mathrm{m}}),} where δ {\displaystyle\delta} istheDiracdeltafunction. CumulativedistributionfunctionParetoTypeIcumulativedistributionfunctionsforvarious α {\displaystyle\alpha} with x m = 1. {\displaystylex_{\mathrm{m}}=1.} Parameters x m > 0 {\displaystylex_{\mathrm{m}}>0} scale(real) α > 0 {\displaystyle\alpha>0} shape(real)Support x ∈ [ x m , ∞ ) {\displaystylex\in[x_{\mathrm{m}},\infty)} PDF α x m α x α + 1 {\displaystyle{\frac{\alphax_{\mathrm{m}}^{\alpha}}{x^{\alpha+1}}}} CDF 1 − ( x m x ) α {\displaystyle1-\left({\frac{x_{\mathrm{m}}}{x}}\right)^{\alpha}} Quantile x m ( 1 − p ) − 1 α {\displaystylex_{\mathrm{m}}{(1-p)}^{-{\frac{1}{\alpha}}}} Mean { ∞ for  α ≤ 1 α x m α − 1 for  α > 1 {\displaystyle{\begin{cases}\infty&{\text{for}}\alpha\leq1\\{\dfrac{\alphax_{\mathrm{m}}}{\alpha-1}}&{\text{for}}\alpha>1\end{cases}}} Median x m 2 α {\displaystylex_{\mathrm{m}}{\sqrt[{\alpha}]{2}}} Mode x m {\displaystylex_{\mathrm{m}}} Variance { ∞ for  α ≤ 2 x m 2 α ( 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}}} Skewness 2 ( 1 + α ) α − 3 α − 2 α  for  α > 3 {\displaystyle{\frac{2(1+\alpha)}{\alpha-3}}{\sqrt{\frac{\alpha-2}{\alpha}}}{\text{for}}\alpha>3} Ex.kurtosis 6 ( α 3 + α 2 − 6 α − 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} Entropy log ⁡ ( ( x m α ) e 1 + 1 α ) {\displaystyle\log\left(\left({\frac{x_{\mathrm{m}}}{\alpha}}\right)\,e^{1+{\tfrac{1}{\alpha}}}\right)} MGF doesnotexistCF α ( − i x m t ) α Γ ( − α , − i x m t ) {\displaystyle\alpha(-ix_{\mathrm{m}}t)^{\alpha}\Gamma(-\alpha,-ix_{\mathrm{m}}t)} Fisherinformation I ( x m , α ) = [ α x m 2 − 1 x m − 1 x m 1 α 2 ] {\displaystyle{\mathcal{I}}(x_{\mathrm{m}},\alpha)={\begin{bmatrix}{\dfrac{\alpha}{x_{\mathrm{m}}^{2}}}&-{\dfrac{1}{x_{\mathrm{m}}}}\\-{\dfrac{1}{x_{\mathrm{m}}}}&{\dfrac{1}{\alpha^{2}}}\end{bmatrix}}} Right: I ( x m , α ) = [ α 2 x m 2 0 0 1 α 2 ] {\displaystyle{\mathcal{I}}(x_{\mathrm{m}},\alpha)={\begin{bmatrix}{\dfrac{\alpha^{2}}{x_{\mathrm{m}}^{2}}}&0\\0&{\dfrac{1}{\alpha^{2}}}\end{bmatrix}}} TheParetodistribution,namedaftertheItaliancivilengineer,economist,andsociologistVilfredoPareto,[1](Italian: [paˈreːto]US:/pəˈreɪtoʊ/pə-RAY-toh),[2]isapower-lawprobabilitydistributionthatisusedindescriptionofsocial,qualitycontrol,scientific,geophysical,actuarial,andmanyothertypesofobservablephenomena.Originallyappliedtodescribingthedistributionofwealthinasociety,fittingthetrendthatalargeportionofwealthisheldbyasmallfractionofthepopulation.[3][4]TheParetoprincipleor"80-20rule"statingthat80%ofoutcomesaredueto20%ofcauseswasnamedinhonourofPareto,buttheconceptsaredistinct,andonlyParetodistributionswithshapevalue(α)of log45 ≈ 1.16preciselyreflectit.Empiricalobservationhasshownthatthis80-20distributionfitsawiderangeofcases,includingnaturalphenomena[5]andhumanactivities.[6][7] Contents 1Definitions 1.1Cumulativedistributionfunction 1.2Probabilitydensityfunction 2Properties 2.1Momentsandcharacteristicfunction 2.2Conditionaldistributions 2.3Acharacterizationtheorem 2.4Geometricmean 2.5Harmonicmean 2.6Graphicalrepresentation 3Relateddistributions 3.1GeneralizedParetodistributions 3.1.1ParetotypesI–IV 3.1.2Feller–Paretodistribution 3.2Relationtotheexponentialdistribution 3.3Relationtothelog-normaldistribution 3.4RelationtothegeneralizedParetodistribution 3.5BoundedParetodistribution 3.5.1GeneratingboundedParetorandomvariables 3.6SymmetricParetodistribution 3.7MultivariateParetodistribution 4Statisticalinference 4.1Estimationofparameters 5Occurrenceandapplications 5.1General 5.2RelationtoZipf'slaw 5.3Relationtothe"Paretoprinciple" 5.4RelationtoPrice'slaw 5.5LorenzcurveandGinicoefficient 6Computationalmethods 6.1Randomsamplegeneration 7Seealso 8References 9Notes 10Externallinks Definitions[edit] IfXisarandomvariablewithaPareto(TypeI)distribution,[8]thentheprobabilitythatXisgreaterthansomenumberx,i.e.thesurvivalfunction(alsocalledtailfunction),isgivenby F ¯ ( x ) = Pr ( X > x ) = { ( x m x ) α x ≥ x m , 1 x < x m , {\displaystyle{\overline{F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac{x_{\mathrm{m}}}{x}}\right)^{\alpha}&x\geqx_{\mathrm{m}},\\1&x 1. {\displaystyle\operatorname{E}(X)={\begin{cases}\infty&\alpha\leq1,\\{\frac{\alphax_{\mathrm{m}}}{\alpha-1}}&\alpha>1.\end{cases}}} ThevarianceofarandomvariablefollowingaParetodistributionis Var ⁡ ( X ) = { ∞ α ∈ ( 1 , 2 ] , ( x m α − 1 ) 2 α α − 2 α > 2. {\displaystyle\operatorname{Var}(X)={\begin{cases}\infty&\alpha\in(1,2],\\\left({\frac{x_{\mathrm{m}}}{\alpha-1}}\right)^{2}{\frac{\alpha}{\alpha-2}}&\alpha>2.\end{cases}}} (Ifα≤1,thevariancedoesnotexist.) Therawmomentsare μ n ′ = { ∞ α ≤ n , α x m n α − n α > n . {\displaystyle\mu_{n}'={\begin{cases}\infty&\alpha\leqn,\\{\frac{\alphax_{\mathrm{m}}^{n}}{\alpha-n}}&\alpha>n.\end{cases}}} Themomentgeneratingfunctionisonlydefinedfornon-positivevaluest ≤ 0as M ( t ; α , x m ) = E ⁡ [ e t X ] = α ( − x m t ) α Γ ( − α , − x m t ) {\displaystyleM\left(t;\alpha,x_{\mathrm{m}}\right)=\operatorname{E}\left[e^{tX}\right]=\alpha(-x_{\mathrm{m}}t)^{\alpha}\Gamma(-\alpha,-x_{\mathrm{m}}t)} M ( 0 , α , x m ) = 1. {\displaystyleM\left(0,\alpha,x_{\mathrm{m}}\right)=1.} Thus,sincetheexpectationdoesnotconvergeonanopenintervalcontaining t = 0 {\displaystylet=0} wesaythatthemomentgeneratingfunctiondoesnotexist. Thecharacteristicfunctionisgivenby φ ( t ; α , x m ) = α ( − i x m t ) α Γ ( − α , − i x m t ) , {\displaystyle\varphi(t;\alpha,x_{\mathrm{m}})=\alpha(-ix_{\mathrm{m}}t)^{\alpha}\Gamma(-\alpha,-ix_{\mathrm{m}}t),} whereΓ(a, x)istheincompletegammafunction. Theparametersmaybesolvedforusingthemethodofmoments.[9] Conditionaldistributions[edit] TheconditionalprobabilitydistributionofaPareto-distributedrandomvariable,giventheeventthatitisgreaterthanorequaltoaparticularnumber  x 1 {\displaystylex_{1}} exceeding x m {\displaystylex_{\text{m}}} ,isaParetodistributionwiththesameParetoindex  α {\displaystyle\alpha} butwithminimum  x 1 {\displaystylex_{1}} insteadof x m {\displaystylex_{\text{m}}} .Thisimpliesthattheconditionalexpectedvalue(ifitisfinite,i.e. α > 1 {\displaystyle\alpha>1} )isproportionalto x 1 {\displaystylex_{1}} .Incaseofrandomvariablesthatdescribethelifetimeofanobject,thismeansthatlifeexpectancyisproportionaltoage,andiscalledtheLindyeffectorLindy'sLaw.[10] Acharacterizationtheorem[edit] Suppose X 1 , X 2 , X 3 , … {\displaystyleX_{1},X_{2},X_{3},\dotsc} areindependentidenticallydistributedrandomvariableswhoseprobabilitydistributionissupportedontheinterval [ x m , ∞ ) {\displaystyle[x_{\text{m}},\infty)} forsome x m > 0 {\displaystylex_{\text{m}}>0} .Supposethatforall n {\displaystylen} ,thetworandomvariables min { X 1 , … , X n } {\displaystyle\min\{X_{1},\dotsc,X_{n}\}} and ( X 1 + ⋯ + X n ) / min { X 1 , … , X n } {\displaystyle(X_{1}+\dotsb+X_{n})/\min\{X_{1},\dotsc,X_{n}\}} areindependent.ThenthecommondistributionisaParetodistribution.[citationneeded] Geometricmean[edit] Thegeometricmean(G)is[11] G = x m exp ⁡ ( 1 α ) . {\displaystyleG=x_{\text{m}}\exp\left({\frac{1}{\alpha}}\right).} Harmonicmean[edit] Theharmonicmean(H)is[11] H = x m ( 1 + 1 α ) . {\displaystyleH=x_{\text{m}}\left(1+{\frac{1}{\alpha}}\right).} Graphicalrepresentation[edit] Thecharacteristiccurved'longtail'distributionwhenplottedonalinearscale,maskstheunderlyingsimplicityofthefunctionwhenplottedonalog-loggraph,whichthentakestheformofastraightlinewithnegativegradient:Itfollowsfromtheformulafortheprobabilitydensityfunctionthatforx≥xm, log ⁡ f X ( x ) = log ⁡ ( α x m α x α + 1 ) = log ⁡ ( α x m α ) − ( α + 1 ) log ⁡ x . {\displaystyle\logf_{X}(x)=\log\left(\alpha{\frac{x_{\mathrm{m}}^{\alpha}}{x^{\alpha+1}}}\right)=\log(\alphax_{\mathrm{m}}^{\alpha})-(\alpha+1)\logx.} Sinceαispositive,thegradient−(α + 1)isnegative. Relateddistributions[edit] GeneralizedParetodistributions[edit] Seealso:GeneralizedParetodistribution Thereisahierarchy[8][12]ofParetodistributionsknownasParetoTypeI,II,III,IV,andFeller–Paretodistributions.[8][12][13]ParetoTypeIVcontainsParetoTypeI–IIIasspecialcases.TheFeller–Pareto[12][14]distributiongeneralizesParetoTypeIV. ParetotypesI–IV[edit] TheParetodistributionhierarchyissummarizedinthenexttablecomparingthesurvivalfunctions(complementaryCDF). Whenμ=0,theParetodistributionTypeIIisalsoknownastheLomaxdistribution.[15] Inthissection,thesymbolxm,usedbeforetoindicatetheminimumvalueofx,isreplacedby σ. Paretodistributions F ¯ ( x ) = 1 − F ( x ) {\displaystyle{\overline{F}}(x)=1-F(x)} Support Parameters TypeI [ x σ ] − α {\displaystyle\left[{\frac{x}{\sigma}}\right]^{-\alpha}} x ≥ σ {\displaystylex\geq\sigma} σ > 0 , α {\displaystyle\sigma>0,\alpha} TypeII [ 1 + x − μ σ ] − α {\displaystyle\left[1+{\frac{x-\mu}{\sigma}}\right]^{-\alpha}} x ≥ μ {\displaystylex\geq\mu} μ ∈ R , σ > 0 , α {\displaystyle\mu\in\mathbb{R},\sigma>0,\alpha} Lomax [ 1 + x σ ] − α {\displaystyle\left[1+{\frac{x}{\sigma}}\right]^{-\alpha}} x ≥ 0 {\displaystylex\geq0} σ > 0 , α {\displaystyle\sigma>0,\alpha} TypeIII [ 1 + ( x − μ σ ) 1 / γ ] − 1 {\displaystyle\left[1+\left({\frac{x-\mu}{\sigma}}\right)^{1/\gamma}\right]^{-1}} x ≥ μ {\displaystylex\geq\mu} μ ∈ R , σ , γ > 0 {\displaystyle\mu\in\mathbb{R},\sigma,\gamma>0} TypeIV [ 1 + ( x − μ σ ) 1 / γ ] − α {\displaystyle\left[1+\left({\frac{x-\mu}{\sigma}}\right)^{1/\gamma}\right]^{-\alpha}} x ≥ μ {\displaystylex\geq\mu} μ ∈ R , σ , γ > 0 , α {\displaystyle\mu\in\mathbb{R},\sigma,\gamma>0,\alpha} Theshapeparameterαisthetailindex,μislocation,σisscale,γisaninequalityparameter.SomespecialcasesofParetoType(IV)are P ( I V ) ( σ , σ , 1 , α ) = P ( I ) ( σ , α ) , {\displaystyleP(IV)(\sigma,\sigma,1,\alpha)=P(I)(\sigma,\alpha),} P ( I V ) ( μ , σ , 1 , α ) = P ( I I ) ( μ , σ , α ) , {\displaystyleP(IV)(\mu,\sigma,1,\alpha)=P(II)(\mu,\sigma,\alpha),} P ( I V ) ( μ , σ , γ , 1 ) = P ( I I I ) ( μ , σ , γ ) . {\displaystyleP(IV)(\mu,\sigma,\gamma,1)=P(III)(\mu,\sigma,\gamma).} Thefinitenessofthemean,andtheexistenceandthefinitenessofthevariancedependonthetailindexα(inequalityindexγ).Inparticular,fractionalδ-momentsarefiniteforsomeδ>0,asshowninthetablebelow,whereδisnotnecessarilyaninteger. MomentsofParetoI–IVdistributions(caseμ=0) E ⁡ [ X ] {\displaystyle\operatorname{E}[X]} Condition E ⁡ [ X δ ] {\displaystyle\operatorname{E}[X^{\delta}]} Condition TypeI σ α α − 1 {\displaystyle{\frac{\sigma\alpha}{\alpha-1}}} α > 1 {\displaystyle\alpha>1} σ δ α α − δ {\displaystyle{\frac{\sigma^{\delta}\alpha}{\alpha-\delta}}} δ < α {\displaystyle\delta 1 {\displaystyle\alpha>1} σ δ Γ ( α − δ ) Γ ( 1 + δ ) Γ ( α ) {\displaystyle{\frac{\sigma^{\delta}\Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}}} 0 < δ < α {\displaystyle0 0 , {\displaystylef(y)={\frac{y^{\gamma_{1}-1}(1-y)^{\gamma_{2}-1}}{B(\gamma_{1},\gamma_{2})}},\qquad00,} whereB( )isthebetafunction.If W = μ + σ ( Y − 1 − 1 ) γ , σ > 0 , γ > 0 , {\displaystyleW=\mu+\sigma(Y^{-1}-1)^{\gamma},\qquad\sigma>0,\gamma>0,} thenWhasaFeller–ParetodistributionFP(μ,σ,γ,γ1,γ2).[8] If U 1 ∼ Γ ( δ 1 , 1 ) {\displaystyleU_{1}\sim\Gamma(\delta_{1},1)} and U 2 ∼ Γ ( δ 2 , 1 ) {\displaystyleU_{2}\sim\Gamma(\delta_{2},1)} areindependentGammavariables,anotherconstructionofaFeller–Pareto(FP)variableis[16] W = μ + σ ( U 1 U 2 ) γ {\displaystyleW=\mu+\sigma\left({\frac{U_{1}}{U_{2}}}\right)^{\gamma}} andwewriteW~FP(μ,σ,γ,δ1,δ2).SpecialcasesoftheFeller–Paretodistributionare F P ( σ , σ , 1 , 1 , α ) = P ( I ) ( σ , α ) {\displaystyleFP(\sigma,\sigma,1,1,\alpha)=P(I)(\sigma,\alpha)} F P ( μ , σ , 1 , 1 , α ) = P ( I I ) ( μ , σ , α ) {\displaystyleFP(\mu,\sigma,1,1,\alpha)=P(II)(\mu,\sigma,\alpha)} F P ( μ , σ , γ , 1 , 1 ) = P ( I I I ) ( μ , σ , γ ) {\displaystyleFP(\mu,\sigma,\gamma,1,1)=P(III)(\mu,\sigma,\gamma)} F P ( μ , σ , γ , 1 , α ) = P ( I V ) ( μ , σ , γ , α ) . {\displaystyleFP(\mu,\sigma,\gamma,1,\alpha)=P(IV)(\mu,\sigma,\gamma,\alpha).} Relationtotheexponentialdistribution[edit] TheParetodistributionisrelatedtotheexponentialdistributionasfollows.IfXisPareto-distributedwithminimumxmandindex α,then Y = log ⁡ ( X x m ) {\displaystyleY=\log\left({\frac{X}{x_{\mathrm{m}}}}\right)} isexponentiallydistributedwithrateparameter α.Equivalently,ifYisexponentiallydistributedwithrate α,then x m e Y {\displaystylex_{\mathrm{m}}e^{Y}} isPareto-distributedwithminimumxmandindex α. Thiscanbeshownusingthestandardchange-of-variabletechniques: Pr ( Y < y ) = Pr ( log ⁡ ( X x m ) < y ) = Pr ( X < x m e y ) = 1 − ( x m x m e y ) α = 1 − e − α y . {\displaystyle{\begin{aligned}\Pr(Y 0 {\displaystyleL>0} location(real) H > L {\displaystyleH>L} location(real) α > 0 {\displaystyle\alpha>0} shape(real)Support L ⩽ x ⩽ H {\displaystyleL\leqslantx\leqslantH} PDF α L α x − α − 1 1 − ( L H ) α {\displaystyle{\frac{\alphaL^{\alpha}x^{-\alpha-1}}{1-\left({\frac{L}{H}}\right)^{\alpha}}}} CDF 1 − L α x − α 1 − ( L H ) α {\displaystyle{\frac{1-L^{\alpha}x^{-\alpha}}{1-\left({\frac{L}{H}}\right)^{\alpha}}}} Mean L α 1 − ( L H ) α ⋅ ( α α − 1 ) ⋅ ( 1 L α − 1 − 1 H α − 1 ) , α ≠ 1 {\displaystyle{\frac{L^{\alpha}}{1-\left({\frac{L}{H}}\right)^{\alpha}}}\cdot\left({\frac{\alpha}{\alpha-1}}\right)\cdot\left({\frac{1}{L^{\alpha-1}}}-{\frac{1}{H^{\alpha-1}}}\right),\alpha\neq1} H L H − L ln ⁡ H L , α = 1 {\displaystyle{\frac{{H}{L}}{{H}-{L}}}\ln{\frac{H}{L}},\alpha=1} Median L ( 1 − 1 2 ( 1 − ( L H ) α ) ) − 1 α {\displaystyleL\left(1-{\frac{1}{2}}\left(1-\left({\frac{L}{H}}\right)^{\alpha}\right)\right)^{-{\frac{1}{\alpha}}}} Variance L α 1 − ( L H ) α ⋅ ( α α − 2 ) ⋅ ( 1 L α − 2 − 1 H α − 2 ) , α ≠ 2 {\displaystyle{\frac{L^{\alpha}}{1-\left({\frac{L}{H}}\right)^{\alpha}}}\cdot\left({\frac{\alpha}{\alpha-2}}\right)\cdot\left({\frac{1}{L^{\alpha-2}}}-{\frac{1}{H^{\alpha-2}}}\right),\alpha\neq2} 2 H 2 L 2 H 2 − L 2 ln ⁡ H L , α = 2 {\displaystyle{\frac{2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln{\frac{H}{L}},\alpha=2} (thisisthesecondrawmoment,notthevariance)Skewness L α 1 − ( L H ) α ⋅ α ∗ ( L k − α − H k − α ) ( α − k ) , α ≠ j {\displaystyle{\frac{L^{\alpha}}{1-\left({\frac{L}{H}}\right)^{\alpha}}}\cdot{\frac{\alpha*(L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}},\alpha\neqj} (thisisthekthrawmoment,nottheskewness) Thebounded(ortruncated)Paretodistributionhasthreeparameters:α,LandH.AsinthestandardParetodistributionαdeterminestheshape.Ldenotestheminimalvalue,andHdenotesthemaximalvalue. Theprobabilitydensityfunctionis α L α x − α − 1 1 − ( L H ) α {\displaystyle{\frac{\alphaL^{\alpha}x^{-\alpha-1}}{1-\left({\frac{L}{H}}\right)^{\alpha}}}} , whereL ≤ x ≤ H,andα > 0. GeneratingboundedParetorandomvariables[edit] IfUisuniformlydistributedon(0, 1),thenapplyinginverse-transformmethod[18] U = 1 − L α x − α 1 − ( L H ) α {\displaystyleU={\frac{1-L^{\alpha}x^{-\alpha}}{1-({\frac{L}{H}})^{\alpha}}}} x = ( − U H α − U L α − H α H α L α ) − 1 α {\displaystylex=\left(-{\frac{UH^{\alpha}-UL^{\alpha}-H^{\alpha}}{H^{\alpha}L^{\alpha}}}\right)^{-{\frac{1}{\alpha}}}} isaboundedPareto-distributed. SymmetricParetodistribution[edit] ThepurposeofSymmetricParetodistributionandZeroSymmetricParetodistributionistocapturesomespecialstatisticaldistributionwithasharpprobabilitypeakandsymmetriclongprobabilitytails.ThesetwodistributionsarederivedfromParetodistribution.Longprobabilitytailnormallymeansthatprobabilitydecaysslowly.Paretodistributionperformsfittingjobinmanycases.Butifthedistributionhassymmetricstructurewithtwoslowdecayingtails,Paretocouldnotdoit.ThenSymmetricParetoorZeroSymmetricParetodistributionisappliedinstead.[19] TheCumulativedistributionfunction(CDF)ofSymmetricParetodistributionisdefinedasfollowing:[19] F ( X ) = P ( x < X ) = { 1 2 ( b 2 b − X ) a X < b 1 − 1 2 ( b X ) a X ≥ b {\displaystyleF(X)=P(x 1. Thereissomenumber0 ≤ p ≤ 1/2suchthat100p %ofallpeoplereceive100(1 − p)%ofallincome,andsimilarlyforeveryreal(notnecessarilyinteger)n > 0,100pn %ofallpeoplereceive100(1 − p)npercentageofallincome.αandparerelatedby 1 − 1 α = ln ⁡ ( 1 − p ) ln ⁡ ( p ) = ln ⁡ ( ( 1 − p ) n ) ln ⁡ ( p n ) {\displaystyle1-{\frac{1}{\alpha}}={\frac{\ln(1-p)}{\ln(p)}}={\frac{\ln((1-p)^{n})}{\ln(p^{n})}}} Thisdoesnotapplyonlytoincome,butalsotowealth,ortoanythingelsethatcanbemodeledbythisdistribution. ThisexcludesParetodistributionsinwhich 0