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| {{Probability distribution
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| | name =Pareto Type I
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| | type =density
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| | pdf_image =[[File:PDF of Pareto Distribution.svg|325px|Pareto Type I probability density functions for various α]]<br /><small>Pareto Type I probability density functions for various α (labeled "k") with ''x''<sub>m</sub> = 1. As α → ∞ the distribution approaches δ(''x'' − ''x''<sub>m</sub>) where δ is the [[Dirac delta function]].</small>
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| | cdf_image =[[File:CDF of Pareto Distribution.svg|325px|Pareto Type I cumulative distribution functions for various α]]<br /><small>Pareto Type I cumulative distribution functions for various α (labeled "k") with ''x''<sub>m</sub> = 1.</small>
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| | parameters =''x''<sub>m</sub> > 0 [[scale parameter|scale]] ([[real number|real]])<br/>α > 0 [[shape parameter|shape]] (real)
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| | support =<math>x \in [x_\mathrm{m}, +\infty)</math>
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| | pdf =<math>\frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m</math>
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| | cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha \text{ for } x \ge x_m</math>
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| | mean =<math>\begin{cases}
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| \infty & \text{for }\alpha\le 1 \\
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| \frac{\alpha\,x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
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| \end{cases}</math>
| |
| | median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math>
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| | mode =<math>x_\mathrm{m}</math>
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| | variance =<math>\begin{cases}
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| \infty & \text{for }\alpha\in(1,2] \\
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| \frac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2
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| \end{cases}</math>
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| | skewness =<math>\frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3</math> | |
| | kurtosis =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4</math>
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| | entropy =<math>\ln\left(\frac{x_\mathrm{m}}{\alpha}\right) + \frac{1}{\alpha} + 1</math>
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| | mgf =<math>\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0</math>
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| | char =<math>\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)</math>
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| | fisher =<math>\begin{pmatrix}\frac{\alpha}{x_m^2} &-\frac{1}{x_m} \\ -\frac{1}{x_m} &\frac{1}{\alpha^2}\end{pmatrix}</math>
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| }}
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| The '''Pareto distribution''', named after the Italian [[civil engineer]] and [[economist]] [[Vilfredo Pareto]], is a [[power law]] [[probability distribution]] that is used in description of [[social sciences|social]], [[scientific]], [[geophysical]], [[actuarial science|actuarial]], and many other types of observable phenomena.
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| == Definition ==
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| If ''X'' is a [[random variable]] with a Pareto (Type I) distribution,<ref name=arnold>{{cite book |author=Barry C. Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn= 0-89974-012-X|ref=harv}}</ref> then the probability that ''X'' is greater than some number ''x'', i.e. the [[survival function]] (also called tail function), is given by
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| | |
| :<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
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| \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
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| 1 & x < x_\mathrm{m}.
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| \end{cases}
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| </math>
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| | |
| where ''x''<sub>m</sub> is the (necessarily positive) minimum possible value of ''X'', and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter ''x''<sub>m</sub> and a shape parameter α, which is known as the ''tail index''. When this distribution is used to model the distribution of wealth, then the parameter α is called the [[Pareto index]].
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| == Properties ==
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| ===Cumulative distribution function===
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| From the definition, the [[cumulative distribution function]] of a Pareto random variable with parameters α and ''x''<sub>m</sub> is
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| | |
| :<math>F_X(x) = \begin{cases}
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| 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
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| 0 & x < x_\mathrm{m}.
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| \end{cases}</math>
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| When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes [[asymptote|asymptotically]]. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a [[log-log plot]], the distribution is represented by a straight line.
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| === Probability density function ===
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| It follows (by [[Derivative|differentiation]]) that the [[probability density function]] is
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| | |
| :<math>f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases} </math>
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| | |
| === Moments and characteristic function ===
| |
| | |
| * The [[expected value]] of a [[random variable]] following a Pareto distribution is
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| :
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| :: <math>E(X)= \begin{cases} \infty & \alpha\le 1, \\
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| \frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha>1.
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| \end{cases}</math>
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| * The [[variance]] of a [[random variable]] following a Pareto distribution is
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| :: <math>\mathrm{Var}(X)= \begin{cases}
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| \infty & \alpha\in(1,2], \\
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| \left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha>2.
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| \end{cases}</math>
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| : (If α ≤ 1, the variance does not exist.)
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| * The raw [[moment (mathematics)|moments]] are
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| :: <math>\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}</math>
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| * The [[Moment-generating function|moment generating function]] is only defined for non-positive values ''t'' ≤ 0 as
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| ::<math>M\left(t;\alpha,x_\mathrm{m}\right) = E \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)</math>
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| ::<math>M\left(0,\alpha,x_\mathrm{m}\right)=1.</math>
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| * The [[Characteristic function (probability theory)|characteristic function]] is given by
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| :: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>
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| : where Γ(''a'', ''x'') is the [[incomplete gamma function]].
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| | |
| === Conditional distributions ===
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| The [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number ''x''<sub>1</sub> exceeding ''x''<sub>m</sub>, is a Pareto distribution with the same Pareto index α but with minimum ''x''<sub>1</sub> instead of ''x''<sub>m</sub>.
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| === A characterization theorem ===
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| Suppose ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... are [[independent identically distributed]] [[random variable]]s whose probability distribution is supported on the interval [''x''<sub>m</sub>, ∞) for some ''x''<sub>m</sub> > 0. Suppose that for all ''n'', the two random variables min{''X''<sub>1</sub>, ..., ''X<sub>n</sub>''} and (''X''<sub>1</sub> + ... + ''X<sub>n</sub>'')/min{''X''<sub>1</sub>, ..., ''X<sub>n</sub>''} are independent. Then the common distribution is a Pareto distribution.{{Citation needed|date=February 2012}}
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| ===Geometric mean===
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| The [[geometric mean]] (''G'') is<ref name=Johnson1994>Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.</ref>
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| : <math> G = x_m \exp \left( \frac{1}{\alpha} \right). </math>
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| ===Harmonic mean===
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| The [[harmonic mean]] (''H'') is<ref name="Johnson1994"/>
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| : <math> H = x_m \left( 1 + \frac{ 1 }{ \alpha } \right) .</math>
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| ==Generalized Pareto distributions==
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| {{See also|Generalized Pareto distribution}}
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| There is a hierarchy <ref name=arnold/><ref name=jkb94>Johnson, Kotz, and Balakrishnan (1994), (20.4).</ref> of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.<ref name=arnold/><ref name=jkb94/><ref name=kk03>{{cite book |author=Christian Kleiber and Samuel Kotz |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=[[John Wiley & Sons|Wiley]] |isbn=0-471-15064-9|ref=harv| url=http://books.google.com/books?id=7wLGjyB128IC&printsec=frontcover}}</ref> Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto<ref name=jkb94/><ref name=feller>{{cite book|last=Feller |first= W.| year=1971| title=An Introduction to Probability Theory and its Applications| volume=II| edition=2nd | location= New York|publisher=Wiley|page=50}} "The densities (4.3) are sometimes called after the economist ''Pareto''. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ ''Ax''<sup>−α</sup> as ''x'' → ∞."</ref> distribution generalizes Pareto Type IV.
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| <!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->
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| ===Pareto types I–IV===
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| The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).<br>
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| When μ = 0, the Pareto distribution Type II is also known as the [[Lomax distribution]].<ref>Lomax, K. S. (1954). Business failures. Another example of the analysis of failure data.''Journal of the American Statistical Association'', 49, 847–852.</ref>
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|
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| In this section, the symbol ''x''<sub>m</sub>, used before to indicate the minimum value of ''x'', is replaced by σ.
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| {| class="wikitable" border="1"
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| |+ Pareto distributions
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| ! !! <math> \overline{F}(x)=1-F(x)</math> !! Support !! Parameters
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| |-
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| | Type I
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| || <math>\left[\frac{x}{\sigma}\right]^{-\alpha}</math>
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| || <math>x > \sigma</math>
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| || <math>\sigma > 0, \alpha</math>
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| |-
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| | Type II
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| || <math>\left[1 + \frac{x-\mu}{\sigma}\right]^{-\alpha}</math>
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| || <math>x > \mu</math>
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| || <math>\mu \in \mathbb R, \sigma > 0, \alpha</math>
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| |-
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| | Lomax
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| || <math>\left[1 + \frac{x}{\sigma}\right]^{-\alpha}</math>
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| || <math>x > 0</math>
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| ||<math>\sigma > 0, \alpha</math>
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| |-
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| | Type III
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| || <math>\left[1 + \left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-1} </math>
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| || <math>x > \mu</math>
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| ||<math> \mu \in \mathbb R, \sigma, \gamma > 0</math>
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| |-
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| | Type IV
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| || <math>\left[1 + \left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-\alpha}</math>
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| || <math>x > \mu</math>
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| || <math>\mu \in \mathbb R, \sigma, \gamma > 0, \alpha</math>
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| |-
| |
| |-
| |
| |}
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| The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are
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| ::<math> P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),</math>
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| ::<math> P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),</math>
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| ::<math> P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).</math>
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| The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.
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| {| class="wikitable" border="1"
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| |+ Moments of Pareto I–IV distributions (case μ = 0)
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| ! !! <math>E[X]</math> !! Condition !! <math>E[X^\delta]</math> !! Condition
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| |-
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| | Type I
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| || <math>\frac{\sigma \alpha}{\alpha-1}</math>
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| || <math>\alpha > 1</math>
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| || <math>\frac{\sigma^\delta \alpha}{\alpha-\delta}</math>
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| || <math> \delta < \alpha</math>
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| |-
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| | Type II
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| || <math> \frac{ \sigma }{\alpha-1}</math>
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| || <math>\alpha > 1</math>
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| || <math> \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}</math>
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| || <math>-1 < \delta < \alpha</math>
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| |-
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| | Type III
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| || <math>\sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)</math>
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| || <math> -1<\gamma<1</math>
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| || <math>\sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)</math>
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| || <math>-\gamma^{-1}<\delta<\gamma^{-1}</math>
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| |-
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| | Type IV
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| || <math>\frac{\sigma\Gamma(\alpha-\gamma \alpha)\Gamma(1+\gamma)}{\Gamma(\alpha)}</math>
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| || <math> -1<\gamma<\alpha</math>
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| || <math>\frac{\sigma^\delta\Gamma(\alpha-\gamma \alpha)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}</math>
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| || <math>-\gamma^{-1}<\delta<\alpha/\gamma </math>
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| |-
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| |-
| |
| |}
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| | |
| ===Feller–Pareto distribution===
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| Feller<ref name=jkb94/><ref name=feller/> defines a Pareto variable by transformation ''U'' = ''Y''<sup>−1</sup> − 1 of a [[beta distribution|beta random variable]] ''Y'', whose probability density function is
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| :<math> f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0<y<1; \gamma_1,\gamma_2>0,</math>
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| where ''B''( ) is the [[beta function]]. If
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| :<math> W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma>0, \gamma>0,</math>
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| then ''W'' has a Feller–Pareto distribution FP(μ, σ, γ, γ<sub>1</sub>, γ<sub>2</sub>).<ref name=arnold/>
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| If <math>U_1 \sim \Gamma(\delta_1, 1)</math> and <math>U_2 \sim \Gamma(\delta_2, 1)</math> are independent [[Gamma distribution|Gamma variables]], another construction of a Feller–Pareto (FP) variable is<ref>{{cite book |last=Chotikapanich |first=Duangkamon |title=Modeling Income Distributions and Lorenz Curves |chapter=Chapter 7: Pareto and Generalized Pareto Distributions |pages=121–122 |url=http://books.google.com/books?id=fUJZZLj1kbwC}}</ref>
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| :<math>W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma</math>
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| | |
| and we write ''W'' ~ FP(μ, σ, γ, δ<sub>1</sub>, δ<sub>2</sub>). Special cases of the Feller–Pareto distribution are
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| :<math>FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)</math>
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| :<math>FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)</math>
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| :<math>FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)</math>
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| :<math>FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).</math>
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| | |
| ==Applications==
| |
| [[Vilfredo Pareto|Pareto]] originally used this distribution to describe the [[Distribution of wealth|allocation of wealth]] among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.<ref>Pareto, Vilfredo, ''Cours d’Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino'', Librairie Droz, Geneva, 1964, pages 299–345.</ref> This idea is sometimes expressed more simply as the [[Pareto principle]] or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.<ref>For a two-quantile population, where approximately 18% of the population owns 82% of the wealth, the [[Theil index]] takes the value 1.</ref> However, the 80-20 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his ''Cours d'économie politique'' indicates that about 30% of the population had about 70% of the income.<!--Fact borrow'd from article "Pareto index"--> The [[probability density function]] (PDF) graph at the beginning of this article shows that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. (Note that the Pareto distribution is not realistic for wealth for the lower end. In fact, [[net worth]] may even be negative.) This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Pareto-distributed:
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| <!-- THESE TWO SEEM TO BELONG UNDER [[Zipf's law]] RATHER THAN THE PARETO DISTRIBUTION
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| * Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)
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| * Frequencies of [[Given_name#Popularity_distribution_of_given_names|given names]] -->
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| * The sizes of human settlements (few cities, many hamlets/villages)<ref name ="Reed">{{cite journal | id = {{citeseerx|10.1.1.70.4555}} | first = William J. | last = Reed | coauthors = et al. | title = The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions | journal = Communications in Statistics – Theory and Methods | volume = 33 | issue = 8 | pages = 1733–1753 | year = 2004. }}</ref>
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| * File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)<ref name ="Reed" />
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| * [[Hard disk drive]] error rates<ref>{{Cite journal |title=Understanding latent sector error and how to protect against them |url=http://www.usenix.org/event/fast10/tech/full_papers/schroeder.pdf |first1=Bianca |last1=Schroeder |first2=Sotirios |last2=Damouras |first3=Phillipa |last3=Gill |journal=8th Usenix Conference on File and Storage Technologies (FAST 2010)| date=2010-02-24 |accessdate=2010-09-10 |quote=We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ<sup>2</sup> statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit. |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| * Clusters of [[Bose–Einstein condensate]] near [[absolute zero]]<ref name="Simon">{{cite journal|last1=Herbert A.|first1=Simon|last2=Ijiri|first2=Yuji|title=Some Distributions Associated with Bose–Einstein Statistics|journal=Proc. Nat. Acad. Sci. USA|date=February 18, 1975|month=May|volume=72|issue=5|pages=1654–1657|accessdate=24 January 2013|pmc=432601|pmid=16578724}}</ref>
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| * The values of [[oil reserves]] in oil fields (a few large fields, many small fields)<ref name ="Reed" />
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| * The length distribution in jobs assigned supercomputers (a few large ones, many small ones){{citation needed|date=December 2010}}
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| * The standardized price returns on individual stocks <ref name ="Reed" />
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| [[File:FitParetoDistr.tif|thumb|240px|Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]
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| * Sizes of sand particles <ref name ="Reed" />
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| * Sizes of meteorites
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| * Numbers of species per genus (There is subjectivity involved: The tendency to divide a genus into two or more increases with the number of species in it){{citation needed|date=December 2010}}
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| * Areas burnt in forest fires
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| * Severity of large [[casualty (person)|casualty]] losses for certain lines of business such as general liability, commercial auto, and workers compensation.<ref>Kleiber and Kotz (2003): page 94.</ref><ref>{{cite journal | last1 = Seal | first1 = H. | year = 1980 | title = Survival probabilities based on Pareto claim distributions | url = | journal = ASTIN Bulletin | volume = 11 | issue = | pages = 61–71 }}</ref>
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| * In [[hydrology]] the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
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| ==Relation to other distributions==
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| === Relation to the exponential distribution ===
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| The Pareto distribution is related to the [[exponential distribution]] as follows. If ''X'' is Pareto-distributed with minimum ''x''<sub>m</sub> and index α, then
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| : <math> Y = \log\left(\frac{X}{x_\mathrm{m}}\right) </math>
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| is [[exponential distribution|exponentially distributed]] with rate parameter α. Equivalently, if ''Y'' is exponentially distributed with rate α, then
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| : <math> x_\mathrm{m} e^Y</math>
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| is Pareto-distributed with minimum ''x''<sub>m</sub> and index α.
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| This can be shown using the standard change of variable techniques:
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| : <math> \Pr(Y<y) = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) = \Pr(X<x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}. </math>
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| The last expression is the cumulative distribution function of an exponential distribution with rate α.
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| ===Relation to the log-normal distribution===
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| Note that the Pareto distribution and [[log-normal distribution]] are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the [[exponential distribution]] and [[normal distribution]].{{citation needed|date=December 2010}}
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| ===Relation to the generalized Pareto distribution===
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| The Pareto distribution is a special case of the [[generalized Pareto distribution]], which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the [[Lomax distribution]] as a special case. This family also contains both the unshifted and shifted [[exponential distribution]]s.
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| === Relation to Zipf's law ===
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| Pareto distributions are continuous probability distributions. [[Zipf's law]], also sometimes called the [[zeta distribution]], may be thought of as a discrete counterpart of the Pareto distribution.
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| === Relation to the "Pareto principle" ===
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| The "[[80-20 law]]", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log<sub>4</sub>(5) = log(5)/log(4), approximately 1.161. This result can be derived from the [[Lorenz curve]] formula given below. Moreover, the following have been shown<ref>{{cite journal | last1 = Hardy | first1 = Michael | year = 2010 | title = Pareto's Law | url = | journal = [[Mathematical Intelligencer]] | volume = 32 | issue = 3| pages = 38–43 | doi = 10.1007/s00283-010-9159-2 }}</ref> to be mathematically equivalent:
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| * Income is distributed according to a Pareto distribution with index α > 1.
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| * There is some number 0 ≤ ''p'' ≤ 1/2 such that 100''p'' % of all people receive 100(1 − ''p'') % of all income, and similarly for every real (not necessarily integer) ''n'' > 0, 100''p<sup>n</sup>'' % of all people receive 100(1 − ''p'')<sup>''n''</sup> percentage of all income.
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| This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
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| This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.
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| == Lorenz curve and Gini coefficient ==
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| [[File:Pareto distributionLorenz.png|thumb|325px|Lorenz curves for a number of Pareto distributions. The case α = ∞ corresponds to perfectly equal distribution (''G'' = 0) and the α = 1 line corresponds to complete inequality (''G'' = 1)]]
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| The [[Lorenz curve]] is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve ''L''(''F'') is written in terms of the PDF ''f'' or the CDF ''F'' as
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| :<math>L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}</math>
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| where ''x''(''F'') is the inverse of the CDF. For the Pareto distribution,
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| :<math>x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}</math>
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| and the Lorenz curve is calculated to be
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| :<math>L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},</math>
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| where α must be greater than or equal to unity, since the denominator in the expression for ''L''(''F'') is just the mean value of ''x''. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
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| The [[Gini coefficient]] is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated to be
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| :<math>G = 1-2 \left (\int_0^1L(F)dF \right ) = \frac{1}{2\alpha-1}</math>
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| (see Aaberge 2005).
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| == Parameter estimation ==
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| The [[likelihood function]] for the Pareto distribution parameters α and ''x''<sub>m</sub>, given a [[sample (statistics)|sample]] ''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>''), is
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| : <math>L(\alpha, x_\mathrm{m}) = \prod _{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod _{i=1}^n \frac {1}{x_i^{\alpha+1}}.</math>
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| Therefore, the logarithmic likelihood function is
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| : <math>\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum _{i=1} ^n \ln x_i.</math>
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| It can be seen that <math>\ell(\alpha, x_\mathrm{m})</math> is monotonically increasing with ''x''<sub>m</sub>, that is, the greater the value of ''x''<sub>m</sub>, the greater the value of the likelihood function. Hence, since ''x'' ≥ ''x''<sub>m</sub>, we conclude that
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| : <math>\widehat x_\mathrm{m} = \min_i {x_i}.</math>
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| To find the [[estimator]] for α, we compute the corresponding partial derivative and determine where it is zero:
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| : <math>\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.</math>
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| Thus the [[maximum likelihood]] estimator for α is:
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| : <math>\widehat \alpha = \frac{n}{\sum _i \left( \ln x_i - \ln \widehat x_\mathrm{m} \right)}.</math>
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| The expected statistical error is:<ref> {{cite journal | author = M. E. J. Newman | year = 2005 | title = Power laws, Pareto distributions and Zipf's law | journal = [[Contemporary Physics]] | volume= 46 | issue = 5 |pages = 323–351| arxiv= cond-mat/0412004 | doi = 10.1080/00107510500052444 |bibcode = 2005ConPh..46..323N }}</ref>
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| : <math>\sigma = \frac {\widehat \alpha} {\sqrt n}. </math>
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| Malik (1970)<ref> {{cite journal | author = H. J. Malik | year = 1970 | title = Estimation of the Parameters of the Pareto Distribution | journal = Metrika | volume= 15 }}</ref> gives the exact joint distribution of <math>(\hat{x}_\mathrm{m},\hat\alpha)</math>. In particular, <math>\hat{x}_\mathrm{m}</math> and <math>\hat\alpha</math> are [[Independence (probability theory)|independent]] and <math>\hat{x}_\mathrm{m}</math> is Pareto with scale parameter ''x''<sub>m</sub> and shape parameter ''n''α, whereas <math>\hat\alpha</math> has an [[Inverse-gamma distribution]] with shape and scale parameters ''n''−1 and ''n''α, respectively.
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| ==Graphical representation==
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| The characteristic curved '[[Long Tail]]' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a [[log-log graph]], which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for ''x'' ≥ ''x''<sub>m</sub>,
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| :<math>\log f_X(x)= \log \left(\alpha\frac{x_\mathrm{m}^\alpha}{x^{\alpha+1}}\right) = \log (\alpha x_\mathrm{m}^\alpha) - (\alpha+1) \log x.</math>
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| Since α is positive, the gradient −(α+1) is negative.
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| == Random sample generation ==
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| Random samples can be generated using [[inverse transform sampling]]. Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] on the unit interval (0, 1], the variate ''T'' given by
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| :<math>T=\frac{x_\mathrm{m}}{U^{\frac{1}{\alpha}}}</math>
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| is Pareto-distributed.<ref>{{cite book |last=Tanizaki |first=Hisashi |title=Computational Methods in Statistics and Econometrics |year=2004 |page=133 |publisher=CRC Press |url=http://books.google.com/books?id=pOGAUcn13fMC&printsec=frontcover}}</ref> If ''U'' is uniformly distributed on [0, 1), it can be exchanged with (1 − ''U'').
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| ==Variants==
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| === Bounded Pareto distribution ===
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| {{See also|Truncated distribution}}
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| {{Probability distribution
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| | name =Bounded Pareto
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| | type =density
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| | pdf_image =
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| | cdf_image =
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| | parameters =
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| <math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
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| <math>H > L</math> [[location parameter|location]] ([[real numbers|real]])<br />
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| <math>\alpha > 0</math> [[shape parameter|shape]] (real)
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| | support =<math>L \leqslant x \leqslant H</math>
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| | pdf =<math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
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| | cdf =<math>\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
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| | mean =<math>\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\neq 1 </math>
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| | median =<math> L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}</math>
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| | mode =
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| | variance =<math>\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\neq 2</math> (this is the second moment, NOT the variance){{Citation needed|date=November 2012}}
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| | skewness = <math>\frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j </math>
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| (this is a formula for the kth moment, NOT the skewness){{Citation needed|date=November 2012}}
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| | kurtosis =
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| | entropy =
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| | mgf =
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| | char =
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| }}
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| The bounded (or truncated) Pareto distribution has three parameters α, ''L'' and ''H''. As in the standard Pareto distribution α determines the shape. ''L'' denotes the minimal value, and ''H'' denotes the maximal value. (The variance in the table on the right should be interpreted as the second moment).
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| The [[probability density function]] is
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| : <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
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| where ''L'' ≤ ''x'' ≤ ''H'', and α > 0.
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| ==== Generating bounded Pareto random variables ====
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| If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0, 1), then applying inverse-transform method <ref>http://www.cs.bgu.ac.il/~mps042/invtransnote.htm</ref>
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| :<math>U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}</math>
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| :<math>x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}</math>
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| is a bounded Pareto-distributed. {{Citation needed|date=February 2011}}
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| {{-}}
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| ===Symmetric Pareto distribution===
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| The symmetric Pareto distribution can be defined by the [[probability density function]]:<ref>{{cite web|title=Do Financial Returns Have Finite or Infinite Variance? A Paradox and an Explanation|author=Grabchak, M. & Samorodnitsky, D.| pages=7–8 |url=http://people.orie.cornell.edu/~gennady/techreports/RetTailParadoxExplFinal.pdf}}</ref>
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| :<math>f(x;\alpha,x_\mathrm{m}) = \begin{cases}
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| \tfrac{1}{2}\alpha x_\mathrm{m}^\alpha |x|^{-\alpha-1} & |x|>x_\mathrm{m} \\
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| 0 & \text{otherwise}.
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| \end{cases}</math>
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| It has a similar shape to a Pareto distribution for ''x'' > ''x''<sub>m</sub> and is [[reflection symmetry|mirror symmetric]] about the vertical axis.
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| ==See also==
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| * [[Bradford's law]]
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| * [[Pareto analysis]]
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| * [[Pareto efficiency]]
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| * [[Pareto interpolation]]
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| * [[Power_law#Power-law_probability_distributions|Power law probability distributions]]
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| * [[Traffic generation model]]
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| ==Notes==
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| {{reflist|30em}}
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| ==References==
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| * {{cite journal |author=M. O. Lorenz |year=1905 |title=Methods of measuring the concentration of wealth |journal=[[Publications of the American Statistical Association]] |volume=9 |issue=70 |pages=209–219 |doi=10.2307/2276207 |bibcode=1905PAmSA...9..209L}}
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| * Pareto V (1965) "La Courbe de la Repartition de la Richesse" (Originally published in 1896). In: Busino G, editor. ''Oevres Completes de Vilfredo Pareto''. Geneva: Librairie Droz. pp. 1–5.
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| * Pareto, V. (1895). La legge della domanda. ''Giornale degli Economisti'', 10, 59–68. English translation in ''Rivista di Politica Economica'', 87 (1997), 691–700.
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| * Pareto, V. (1897). ''Cours d'économie politique''. Lausanne: Ed. Rouge.
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| ==External links==
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| * Gini's Nuclear Family / Rolf Aabergé. – In: [http://www.unisi.it/eventi/GiniLorenz05/ International Conference to Honor Two Eminent Social Scientists], May, 2005 – [http://www.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf PDF]
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| * {{springer|title=Pareto distribution|id=p/p071580}}
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| * [http://www.csee.usf.edu/~christen/tools/syntraf1.c syntraf1.c] is a C program to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.
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| * [http://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf "Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes" /Mark E. Crovella and Azer Bestavros ]
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| * {{MathWorld |title=Pareto distribution |id=ParetoDistribution}}
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Pareto Distribution}}
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| [[Category:Actuarial science]]
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| [[Category:Continuous distributions]]
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| [[Category:Power laws]]
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| [[Category:Socioeconomics]]
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| [[Category:Probability distributions with non-finite variance]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions]]
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