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}}