Editing Pareto distribution

{{short description|Probability distribution}}
{{CS1 config|mode=cs1}}
{{Infobox probability distribution
 | name       =Pareto Type I
 | type       =density
 | pdf_image  =[[File:Probability density function of Pareto distribution.svg|325px|Pareto Type I probability density functions for various ''α'']]<br />Pareto Type I probability density functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> As <math>\alpha \rightarrow \infty,</math> the distribution approaches <math>\delta(x - x_\mathrm{m}),</math> where <math>\delta</math> is the [[Dirac delta function]].
 | cdf_image  =[[File:Cumulative distribution function of Pareto distribution.svg|325px|Pareto Type I cumulative distribution functions for various ''α'']]<br />Pareto Type I cumulative distribution functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math>
 | parameters =<math>x_\mathrm{m} > 0</math> [[Scale parameter|scale]] ([[real number|real]])<br /><math>\alpha > 0</math> [[Shape parameter|shape]] (real)
 | support    =<math>x \in [x_\mathrm{m}, \infty)</math>
 | pdf        =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math>
 | cdf        =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math>
 | quantile   = <math>x_\mathrm{m} {(1 - p)}^{-\frac{1}{\alpha}}</math>
 | mean       =<math>\begin{cases}
     \infty & \text{for }\alpha\le 1 \\
     \dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
   \end{cases}</math>
 | median     =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math>
 | mode       =<math>x_\mathrm{m}</math>
 | variance   =<math>\begin{cases}
     \infty & \text{for }\alpha\le 2 \\
     \dfrac{x_\mathrm{m}^2\alpha}{(\alpha- 1)^2(\alpha-2)} & \text{for }\alpha>2
   \end{cases}</math>
 | 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>
 | entropy    =<math>\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right) </math>
 | mgf        =does not exist
 | char       =<math>\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)</math>
 | fisher     =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
                   \dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
                   0 & \dfrac{1}{\alpha^2}
               \end{bmatrix}</math>
 | ES       =<math>\frac{ x_m \alpha }{ (1-p)^{\frac{1}{\alpha}}  (\alpha-1)}</math><ref name="norton">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer|doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |s2cid=254231768 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27}}</ref>
 | bPOE     =<math>\left( \frac{x_m \alpha}{x(\alpha-1) }  \right)^\alpha </math><ref name="norton"/>
}}

The '''Pareto distribution''', named after the Italian [[civil engineer]], [[economist]], and [[sociologist]] [[Vilfredo Pareto]],<ref>{{cite journal |last=Amoroso |first=Luigi|date=January 1938 |title=Vilfredo Pareto |journal=Econometrica (Pre-1986) |via=ProQuest |volume=6 |issue=1 }}</ref> is a [[power-law]] [[probability distribution]] that is used in description of [[social]], [[quality control]], [[scientific]], [[geophysical]], [[actuarial science|actuarial]], and many other types of observable phenomena; the principle 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 population.<ref>{{cite journal |last=Pareto |first=Vilfredo |year=1898 |title=Cours d'economie politique |journal=Journal of Political Economy |volume=6 |doi=10.1086/250536 |url=https://zenodo.org/record/2144014 }}</ref><ref name=":1"/>

The ''[[Pareto principle]]'' or "80:20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value ({{math|''&alpha;''}}) {{nobr|of &hairsp; {{math|log&hairsp;{{sub|4}}&thinsp;5 ≈ 1.16}}}} precisely reflect it. Empirical observation has shown that this 80:20 distribution fits a wide range of cases, including natural phenomena<ref>{{cite journal |last=van Montfort |first=M.A.J. |year=1986 |title=The generalized Pareto distribution applied to rainfall depths |journal=Hydrological Sciences Journal |volume=31 |issue=2 |pages=151–162 |doi=10.1080/02626668609491037 |doi-access=free |bibcode=1986HydSJ..31..151V }}</ref> and human activities.<ref>{{cite journal |last=Oancea |first=Bogdan |year=2017 |title=Income inequality in Romania: The exponential-Pareto distribution |journal=Physica A: Statistical Mechanics and Its Applications |volume=469 |pages=486–498 |doi=10.1016/j.physa.2016.11.094 |bibcode=2017PhyA..469..486O }}</ref><ref>{{cite web |last=Morella |first=Matteo |title=Pareto distribution |url=https://www.academia.edu/59302211 |website=academia.edu}}</ref>

==Definitions==
If ''X'' is a [[random variable]] with a Pareto (Type I) distribution,<ref name=arnold>{{cite book |first=Barry C. |last=Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn= 978-0-89974-012-6}}</ref> then the probability that ''X'' is greater than some number ''x'', i.e., the [[survival function]] (also called tail function), is given by

:<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
1 & x < x_\mathrm{m},
\end{cases}
</math>

where ''x''<sub>m</sub> is the (necessarily positive) minimum possible value of ''X'', and ''α'' is a positive parameter. The type I Pareto distribution is characterized by a [[scale parameter]] ''x''<sub>m</sub> and a [[shape parameter]] ''α'', which is known as the ''tail index''. If this distribution is used to model the distribution of wealth, then the parameter ''α'' is called the [[Pareto index]].

===Cumulative distribution function===
From the definition, the [[cumulative distribution function]] of a Pareto random variable with parameters ''α'' and ''x''<sub>m</sub> is

:<math>F_X(x) = \begin{cases}
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
0 & x < x_\mathrm{m}.
\end{cases}</math>

===Probability density function===
It follows (by [[Derivative|differentiation]]) that the [[probability density function]] is

:<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>

When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes [[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.

==Properties==
===Moments and characteristic function===
* The [[expected value]] of a [[random variable]] following a Pareto distribution is
:
:: <math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
\frac{\alpha x_{\mathrm{m}}}{\alpha-1} & \alpha>1.
\end{cases}</math>
* The [[variance]] of a [[random variable]] following a Pareto distribution is

:: <math>\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}</math>

: (If ''α'' ≤ 2, the variance does not exist.)
* The raw [[moment (mathematics)|moments]] are

:: <math>\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}</math>
* The [[Moment-generating function|moment generating function]] is only defined for non-positive values ''t''&nbsp;≤&nbsp;0 as

::<math>M\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)</math>
::<math>M\left(0,\alpha,x_\mathrm{m}\right)=1.</math>
Thus, since the expectation does not converge on an [[open interval]] containing <math>t=0</math> we say that the moment generating function does not exist.
* The [[Characteristic function (probability theory)|characteristic function]] is given by

:: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>

: where Γ(''a'',&nbsp;''x'') is the [[incomplete gamma function]].

The parameters may be solved for using the [[Method of moments (statistics)|method of moments]].<ref>S. Hussain, S.H. Bhatti (2018).  
[https://www.researchgate.net/publication/322758024_Parameter_estimation_of_Pareto_distribution_Some_modified_moment_estimators Parameter estimation of Pareto distribution: Some modified moment estimators].  ''Maejo International Journal of Science and Technology'' 12(1):11-27.</ref>

===Conditional distributions===
The [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number&nbsp;<math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index&nbsp;<math>\alpha</math> but with minimum&nbsp;<math>x_1</math> instead of <math>x_\text{m}</math>:

:<math>
\text{Pr}(X \geq x | X \geq x_1) = 
\begin{cases} 
\left(\frac{x_1}{x}\right)^\alpha & x \geq x_1, \\
1 & x < x_1.
\end{cases}
</math>

This implies that the conditional expected value (if it is finite, i.e. <math>\alpha>1</math>) is proportional to <math>x_1</math>:

:<math>\text{E}(X | X \geq x_1) \propto x_1.</math>

In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called the [[Lindy effect]] or Lindy's Law.<ref name=":02">{{cite journal|last1=Eliazar|first1=Iddo|date=November 2017|title=Lindy's Law|journal=Physica A: Statistical Mechanics and Its Applications|volume=486|pages=797–805|bibcode=2017PhyA..486..797E|doi=10.1016/j.physa.2017.05.077|s2cid=125349686 }}</ref>

===A characterization theorem===
Suppose <math>X_1, X_2, X_3, \dotsc</math> are [[independent identically distributed]] [[random variable]]s whose probability distribution is supported on the interval <math>[x_\text{m},\infty)</math> for some <math>x_\text{m}>0</math>. Suppose that for all <math>n</math>, the two random variables <math>\min\{X_1,\dotsc,X_n\}</math> and <math>(X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\}</math> are independent. Then the common distribution is a Pareto distribution.{{Citation needed|date=February 2012}}

===Geometric mean===
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>

: <math> G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).</math>

===Harmonic mean===
The [[harmonic mean]] (''H'') is<ref name="Johnson1994"/>

: <math> H = x_\text{m} \left( 1 + \frac{ 1 }{ \alpha } \right).</math>

===Graphical representation===
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>,

:<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>

Since ''α'' is positive, the gradient −(''α''&nbsp;+&nbsp;1) is negative.

==Related distributions==
===Generalized Pareto distributions===
{{See also|Generalized Pareto distribution}}

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 |author1=Christian Kleiber  |author2=Samuel Kotz  |name-list-style=amp |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=[[John Wiley & Sons|Wiley]] |isbn=978-0-471-15064-0| url=https://books.google.com/books?id=7wLGjyB128IC}}</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''&nbsp;→&nbsp;∞".</ref> distribution generalizes Pareto Type IV.
<!--- In this context using x_m for the lower bound for the scale parameter is not meaningful, usual notation is \sigma --->

====Pareto types I–IV====
The Pareto distribution hierarchy is summarized in the next table comparing the [[survival function]]s (complementary CDF).

When ''μ'' = 0, the Pareto distribution Type II is also known as the [[Lomax distribution]].<ref>{{cite journal | last1 = Lomax | first1 = K. S. | year = 1954 | title = Business failures. Another example of the analysis of failure data | journal = Journal of the American Statistical Association | volume = 49 | issue = 268| pages = 847–52 | doi=10.1080/01621459.1954.10501239}}</ref>

In this section, the symbol ''x''<sub>m</sub>, used before to indicate the minimum value of ''x'', is replaced by&nbsp;''σ''.

{|class="wikitable"
|+Pareto distributions
! !! <math> \overline{F}(x)=1-F(x)</math> !! Support !! Parameters
|-
| Type I
|| <math>\left[\frac x \sigma \right]^{-\alpha}</math>
|| <math>x \ge \sigma</math>
|| <math>\sigma > 0, \alpha</math>
|-
| Type II
|| <math>\left[1 + \frac{x-\mu} \sigma \right]^{-\alpha}</math>
|| <math>x \ge \mu</math>
|| <math>\mu \in \mathbb R, \sigma > 0, \alpha</math>
|-
| Lomax
|| <math>\left[1 + \frac x \sigma \right]^{-\alpha}</math>
|| <math>x \ge 0</math>
|| <math>\sigma > 0, \alpha</math>
|-
| Type III
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1} </math>
|| <math>x \ge \mu</math>
|| <math> \mu \in \mathbb R, \sigma, \gamma > 0</math>
|-
| Type IV
|| <math>\left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha}</math>
|| <math>x \ge \mu</math>
|| <math>\mu \in \mathbb R, \sigma, \gamma > 0, \alpha</math>
|-
|-
|}

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

::<math> P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),</math>
::<math> P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),</math>
::<math> P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).</math>

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.

{|class="wikitable"
|+Moments of Pareto I–IV distributions (case ''μ'' = 0)
! !! <math>\operatorname{E}[X]</math> !! Condition !! <math>\operatorname{E}[X^\delta]</math> !! Condition
|-
| Type I
|| <math>\frac{\sigma \alpha}{\alpha-1}</math>
|| <math>\alpha > 1</math>
|| <math>\frac{\sigma^\delta \alpha}{\alpha-\delta}</math>
|| <math> \delta < \alpha</math>
|-
| Type II
|| <math> \frac{ \sigma }{\alpha-1}+\mu</math>
|| <math>\alpha > 1</math>
|| <math> \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}</math>
|| <math>0 < \delta < \alpha</math>
|-
| Type III
|| <math>\sigma\Gamma(1-\gamma)\Gamma(1 + \gamma)</math>
|| <math> -1<\gamma<1</math>
|| <math>\sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta)</math>
|| <math>-\gamma^{-1}<\delta<\gamma^{-1}</math>
|-
| Type IV
|| <math>\frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)}</math>
|| <math> -1<\gamma<\alpha</math>
|| <math>\frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}</math>
|| <math>-\gamma^{-1}<\delta<\alpha/\gamma </math>
|-
|-
|}

====Feller–Pareto distribution====
Feller<ref name=jkb94/><ref name=feller/> defines a Pareto variable by transformation ''U''&nbsp;=&nbsp;''Y''<sup>−1</sup>&nbsp;−&nbsp;1 of a [[beta distribution|beta random variable]] ,''Y'', whose probability density function is

:<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>

where ''B''(&nbsp;) is the [[beta function]]. If

:<math> W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma>0, \gamma>0,</math>

then ''W'' has a Feller–Pareto distribution FP(''μ'', ''σ'', ''γ'', ''γ''<sub>1</sub>, ''γ''<sub>2</sub>).<ref name=arnold/>

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 |date=16 September 2008 |pages=121–22 |publisher=Springer |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>

:<math>W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma</math>

and we write ''W'' ~ FP(''μ'', ''σ'', ''γ'', ''δ''<sub>1</sub>, ''δ''<sub>2</sub>). Special cases of the Feller–Pareto distribution are

:<math>FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)</math>
:<math>FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)</math>
:<math>FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)</math>
:<math>FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).</math>

===Inverse-Pareto Distribution / Power Distribution ===

When a random variable <math>Y</math> follows a pareto distribution, then its inverse <math>X=1/Y</math> follows a Power distribution. 
Inverse Pareto distribution is equivalent to a Power distribution <ref>Dallas, A. C. "Characterizing the Pareto and power distributions." Annals of the Institute of Statistical Mathematics 28.1 (1976): 491-497.</ref>

:<math>Y\sim \mathrm{Pa}(\alpha, x_m) = \frac{\alpha x_m^\alpha}{y^{\alpha+1}} \quad (y \ge x_m) \quad \Leftrightarrow \quad X\sim \mathrm{iPa}(\alpha, x_m) = \mathrm{Power}(x_m^{-1}, \alpha) = \frac{\alpha x^{\alpha-1}}{(x_m^{-1})^\alpha} \quad (0< x \le x_m^{-1})</math>

===Relation to the exponential distribution===
The Pareto distribution is related to the [[exponential distribution]] as follows. If ''X'' is Pareto-distributed with minimum ''x''<sub>m</sub> and index&nbsp;''α'', then

: <math> Y = \log\left(\frac{X}{x_\mathrm{m}}\right) </math>

is [[exponential distribution|exponentially distributed]] with rate parameter&nbsp;''α''. Equivalently, if ''Y'' is exponentially distributed with rate&nbsp;''α'', then

: <math> x_\mathrm{m} e^Y</math>

is Pareto-distributed with minimum ''x''<sub>m</sub> and index&nbsp;''α''.

This can be shown using the standard change-of-variable techniques:

: <math>
\begin{align}
\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}.
\end{align}
</math>

The last expression is the cumulative distribution function of an exponential distribution with rate&nbsp;''α''.

Pareto distribution can be constructed by hierarchical exponential distributions.<ref>{{Cite thesis|title=Bayesian semiparametric spatial and joint spatio-temporal modeling|url=https://mospace.umsystem.edu/xmlui/handle/10355/4450|publisher=University of Missouri--Columbia|date=2006|degree=Thesis|first=Gentry|last=White}} section 5.3.1.</ref> Let
<math>\phi | a \sim \text{Exp}(a)</math> and
<math>\eta | \phi \sim \text{Exp}(\phi) </math>. Then we have <math>p(\eta | a) = \frac{a}{(a+\eta)^2}</math> and, as a result, <math>a+\eta \sim \text{Pareto}(a, 1)</math>.

More in general, if <math>\lambda \sim \text{Gamma}(\alpha, \beta)</math> (shape-rate parametrization) and <math>\eta | \lambda \sim \text{Exp}(\lambda) </math>, then <math>\beta + \eta \sim \text{Pareto}(\beta, \alpha)</math>.

Equivalently, if <math>Y \sim \text{Gamma}(\alpha,1) </math> and <math>X \sim \text{Exp}(1)</math>, then <math>x_{\text{m}} \! \left(1 + \frac{X}{Y}\right) \sim \text{Pareto}(x_{\text{m}}, \alpha)</math>.

===Relation to the log-normal distribution===
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]]. (See [[#Relation_to_the_exponential_distribution|the previous section]].)

===Relation to the generalized Pareto distribution===
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.

The Pareto distribution with scale <math>x_m</math> and shape <math>\alpha</math> is equivalent to the generalized Pareto distribution with location <math>\mu=x_m</math>, scale <math>\sigma=x_m/\alpha</math> and shape <math>\xi=1/\alpha</math>  and, conversely, one can get the Pareto distribution from the GPD by taking <math>x_m = \sigma/\xi</math> and <math>\alpha=1/\xi</math> if <math>\xi > 0</math>.

===Bounded Pareto distribution===
{{See also|Truncated distribution}}
{{Probability distribution
 | name       =Bounded Pareto
 | type       =density
 | pdf_image  =
 | cdf_image  =
 | parameters =
<math>L > 0</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>H > L</math> [[location parameter|location]] ([[real numbers|real]])<br />
<math>\alpha > 0</math> [[shape parameter|shape]] (real)
 | support    =<math>L \leqslant x \leqslant H</math>
 | pdf        =<math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
 | cdf        =<math>\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha}</math>
 | 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><br />
<math>\frac{{H}{L}}{{H}-{L}}\ln\frac{H}{L}, \alpha=1</math>
 | median     =<math> L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}</math>
 | mode       =
 | 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>
<math>\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2</math>
(this is the second raw moment, not the variance)
 | 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>
(this is the kth raw moment, not the skewness)
 | kurtosis   =
 | entropy    =
 | mgf        =
 | char       =
}}

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 [[probability density function]] is

: <math>\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha}</math>,

where ''L''&nbsp;≤&nbsp;''x''&nbsp;≤&nbsp;''H'', and ''α''&nbsp;>&nbsp;0.

====Generating bounded Pareto random variables====
If ''U'' is [[uniform distribution (continuous)|uniformly distributed]] on (0,&nbsp;1), then applying inverse-transform method <ref>{{Cite web |url=http://www.cs.bgu.ac.il/~mps042/invtransnote.htm |title=Inverse Transform Method |access-date=2012-08-27 |archive-date=2012-01-17 |archive-url=https://web.archive.org/web/20120117042753/http://www.cs.bgu.ac.il/~mps042/invtransnote.htm |url-status=dead }}</ref>

:<math>U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}</math>
:<math>x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}</math>

is a bounded Pareto-distributed.
{{Clear}}

===Symmetric Pareto distribution===
The purpose of the Symmetric and Zero Symmetric Pareto distributions is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from the Pareto distribution. Long probability tails normally means that probability decays slowly, and can be used to fit a variety of datasets. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.<ref name=":0">{{Cite journal|last=Huang|first=Xiao-dong|date=2004|title=A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic|journal=IEEE Transactions on Broadcasting|volume=50|issue=3|pages=323–334|doi=10.1109/TBC.2004.834013}}</ref>

The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:<ref name=":0" />

<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over 2b-X}) ^a  & X<b \\
1- \tfrac{1}{2}(\tfrac{b}{X})^a& X\geq b
\end{cases}</math>

The corresponding probability density function (PDF) is:<ref name=":0" />

<math>p(x) = {ab^a \over 2(b+\left\vert x-b \right\vert)^{a+1}},X\in R</math>

This distribution has two parameters: a and b. It is symmetric about b. Then the mathematic expectation is b. When, it has variance as following:

<math>E((x-b)^2)=\int_{-\infty}^{\infty} (x-b)^2p(x)dx={2b^2 \over (a-2)(a-1) }

</math>

The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:

<math>F(X) = P(x < X ) = \begin{cases}
\tfrac{1}{2}({b \over b-X}) ^a  & X<0 \\
1- \tfrac{1}{2}(\tfrac{b}{b+X})^a& X\geq 0
\end{cases}</math>

The corresponding PDF is:

<math>p(x) = {ab^a \over 2(b+\left\vert x \right\vert)^{a+1}},X\in R</math>

This distribution is symmetric about zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.<ref name=":0" />

===Multivariate Pareto distribution===
The univariate Pareto distribution has been extended to a [[multivariate Pareto distribution]].<ref>{{cite journal
|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader 
|title=Multivariate generalized Pareto distributions |journal=Bernoulli|volume=12|year=2006|number=5 |pages=917–30 
|doi=10.3150/bj/1161614952 
|citeseerx=10.1.1.145.2991|s2cid=16504396 }}</ref>

==Statistical inference==

===Estimation of parameters===
The [[likelihood function]] for the Pareto distribution parameters ''α'' and ''x''<sub>m</sub>, given an independent [[sample (statistics)|sample]] ''x'' =&nbsp;(''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x<sub>n</sub>''), is

: <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>

Therefore, the logarithmic likelihood function is

: <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>

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

: <math>\widehat x_\mathrm{m} = \min_i {x_i}.</math>

To find the [[estimator]] for ''α'', we compute the corresponding partial derivative and determine where it is zero:

: <math>\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.</math>

Thus the [[maximum likelihood]] estimator for ''α'' is:

: <math>\widehat \alpha = \frac{n}{\sum _i  \ln (x_i/\widehat x_\mathrm{m}) }.</math>

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=323–51| arxiv=cond-mat/0412004 |doi=10.1080/00107510500052444 |bibcode=2005ConPh..46..323N|s2cid=202719165 }}</ref>

: <math>\sigma = \frac {\widehat \alpha} {\sqrt n}. </math>

Malik (1970)<ref>{{cite journal |author=H. J. Malik |year=1970 |title=Estimation of the Parameters of the Pareto Distribution |journal=Metrika |volume=15|pages=126–132 |doi=10.1007/BF02613565 |s2cid=124007966 }}</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 ''nα'', whereas <math>\hat\alpha</math> has an [[inverse-gamma distribution]] with shape and scale parameters ''n''&nbsp;−&nbsp;1 and ''nα'', respectively.

==Occurrence and applications==
===General===
[[Vilfredo 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 name=":1">Pareto, Vilfredo, ''Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino'', Librairie Droz, Geneva, 1964, pp. 299–345. [https://web.archive.org/web/20130531151249/http://www.institutcoppet.org/wp-content/uploads/2012/05/Cours-d%C3%A9conomie-politique-Tome-II-Vilfredo-Pareto.pdf Original book archived]</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> As Michael Hudson points out (''The Collapse of Antiquity'' [2023] p. 85 & n.7) "a mathematical corollary [is] that 10% would have 65% of the wealth, and 5% would have half the national wealth.” 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.{{citation needed|date=May 2019}} 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. (The Pareto distribution is not realistic for wealth for the lower end, however. 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:
<!-- THESE TWO SEEM TO BELONG UNDER [[Zipf's law]] RATHER THAN THE PARETO DISTRIBUTION
* Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)
* Frequencies of [[Given name#Popularity distribution of given names|given names]] -->
* All four variables of the household's budget constraint: consumption, labor income, capital income, and wealth.<ref>{{cite web |ssrn=4636704 |last1=Gaillard |first1=Alexandre |last2=Hellwig |first2=Christian |last3=Wangner | first3=Philipp |last4=Werquin |first4=Nicolas |title=Consumption, Wealth, and Income Inequality: A Tale of Tails |date=2023  |doi=10.2139/ssrn.4636704 | url=https://ssrn.com/abstract=4636704 }}</ref>
* The sizes of human settlements (few cities, many hamlets/villages)<ref name="Reed">{{cite journal |citeseerx=10.1.1.70.4555 |first=William J. |last=Reed |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=1733–53 |year=2004 |doi=10.1081/sta-120037438|s2cid=13906086 |display-authors=etal}}</ref><ref name="Reed2002">{{cite journal |first=William J. |last=Reed |title=On the rank-size distribution for human settlements |journal=Journal of Regional Science |volume=42 |issue=1 |pages=1–17 |year=2002 |doi=10.1111/1467-9787.00247|bibcode=2002JRegS..42....1R |s2cid=154285730 }}</ref>
* File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)<ref name ="Reed" />
* [[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 |author1-link= Bianca Schroeder |first2=Sotirios |last2=Damouras |first3=Phillipa |last3=Gill |journal=8th Usenix Conference on File and Storage Technologies (FAST 2010)| date=2010-02-24 |access-date=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.}}</ref>
* Clusters of [[Bose–Einstein condensate]] near [[absolute zero]]<ref name="Simon">{{cite journal|first2=Herbert A.|last2=Simon|author=Yuji Ijiri |title=Some Distributions Associated with Bose–Einstein Statistics|journal=Proc. Natl. Acad. Sci. USA|date=May 1975|volume=72|issue=5|pages=1654–57|pmc=432601|pmid=16578724|doi=10.1073/pnas.72.5.1654|bibcode=1975PNAS...72.1654I|doi-access=free}}</ref>
[[File:FitParetoDistr.tif|thumb|250px|Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]
* The values of [[oil reserves]] in oil fields (a few [[Giant oil and gas fields|large fields]], many [[Stripper well|small fields]])<ref name ="Reed" />
* The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)<ref>{{Cite journal|last1=Harchol-Balter|first1=Mor|author1-link=Mor Harchol-Balter|last2=Downey|first2=Allen|date=August 1997|title=Exploiting Process Lifetime Distributions for Dynamic Load Balancing|url=https://users.soe.ucsc.edu/~scott/courses/Fall11/221/Papers/Sync/harcholbalter-tocs97.pdf|journal=ACM Transactions on Computer Systems|volume=15|issue=3|pages=253–258|doi=10.1145/263326.263344|s2cid=52861447}}</ref>
* The standardized price returns on individual stocks <ref name="Reed" />
* Sizes of sand particles <ref name ="Reed" />
* The size of meteorites
* 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): p. 94.</ref><ref>{{cite journal |last1=Seal |first1=H. |year=1980 |title=Survival probabilities based on Pareto claim distributions |journal=ASTIN Bulletin |volume=11 |pages=61–71|doi=10.1017/S0515036100006620 |doi-access=free }}</ref>
* Amount of time a user on [[Steam (service)|Steam]] will spend playing different games. (Some games get played a lot, but most get played  almost never.) [https://docs.google.com/spreadsheets/d/1AjtfgTQc1T84NCyJWGcCPN4jrVsOpX0bp0jgPZJEW6A/edit#gid=0]{{Original research inline|date=December 2020}}
* In [[hydrology]] the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.<ref>CumFreq, software for cumulative frequency analysis and probability distribution fitting [https://www.waterlog.info/cumfreq.htm]</ref> 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]].
* In Electric Utility Distribution Reliability (80% of the Customer Minutes Interrupted occur on approximately 20% of the days in a given year).

===Relation to Zipf's law===
The Pareto distribution is a continuous probability distribution. [[Zipf's law]], also sometimes called the [[zeta distribution]], is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1.  Zipf's can be derived from the Pareto distribution if the <math>x</math> values (incomes) are binned into <math>N</math> ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining <math>x_m</math> so that <math>\alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)}</math> where <math>H(N,\alpha-1)</math> is the [[Harmonic number#Generalized harmonic numbers|generalized harmonic number]]. This makes Zipf's probability density function derivable from Pareto's.

: <math>f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)}</math>

where  <math>s = \alpha-1</math> and <math>x</math> is an integer representing rank from 1 to N where N is the highest income bracket.  So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has <math>f(x)</math> probability of ranking <math>x</math>.

===Relation to the "Pareto principle"===
The "[[Pareto principle|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 <math>\alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161</math>. 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 |journal=[[Mathematical Intelligencer]] |volume=32 |issue=3 |pages=38–43 |doi=10.1007/s00283-010-9159-2|s2cid=121797873 }}</ref> to be mathematically equivalent:
* Income is distributed according to a Pareto distribution with index ''α''&nbsp;>&nbsp;1.
* There is some number 0&nbsp;≤&nbsp;''p''&nbsp;≤&nbsp;1/2 such that 100''p'' % of all people receive 100(1&nbsp;−&nbsp;''p'')% of all income, and similarly for every real (not necessarily integer) ''n''&nbsp;>&nbsp;0, 100''p<sup>n</sup>'' % of all people receive 100(1&nbsp;−&nbsp;''p'')<sup>''n''</sup> percentage of all income. ''α'' and ''p'' are related by
:: <math>1-\frac{1}{\alpha}=\frac{\ln(1-p)}{\ln(p)}=\frac{\ln((1-p)^n)}{\ln(p^n)}</math>

This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.

This excludes Pareto distributions in which&nbsp;0&nbsp;<&nbsp;''α''&nbsp;≤&nbsp;1, which, as noted above, have an infinite expected value, and so cannot reasonably model income distribution.

===Relation to Price's law===
[[Price's law]] is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that <math>\alpha=1</math>. Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.

===Lorenz curve and Gini coefficient===
[[File:ParetoLorenzSVG.svg|thumb|325px|Lorenz curves for a number of Pareto distributions. The case ''α''&nbsp;=&nbsp;∞ corresponds to perfectly equal distribution (''G''&nbsp;=&nbsp;0) and the ''α''&nbsp;=&nbsp;1 line corresponds to complete inequality (''G''&nbsp;=&nbsp;1)]]

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

:<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>

where ''x''(''F'') is the inverse of the CDF. For the Pareto distribution,

:<math>x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}}</math>

and the Lorenz curve is calculated to be

:<math>L(F) = 1-(1-F)^{1-\frac{1}{\alpha}},</math>

For <math>0<\alpha\le 1</math> the denominator is infinite, yielding ''L''=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.

According to [[Oxfam]] (2016) the richest 62 people have as much wealth as the poorest half of the world's population.<ref>{{cite web|title=62 people own the same as half the world, reveals Oxfam Davos report|url=https://www.oxfam.org/en/pressroom/pressreleases/2016-01-18/62-people-own-same-half-world-reveals-oxfam-davos-report|publisher=Oxfam|date=Jan 2016}}</ref> We can estimate the Pareto index that would apply to this situation. Letting ε equal <math>62/(7\times 10^9)</math> we have:
:<math>L(1/2)=1-L(1-\varepsilon)</math>
or
:<math>1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}}</math>
<!--:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(1-\frac{1}{\alpha})\ln\varepsilon</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=(\ln\varepsilon/\ln 2)(1-\frac{1}{\alpha})\ln 2</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})</math>
:<math>\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx(\ln\varepsilon/\ln 2)(1-(1/2)^{1-\frac{1}{\alpha}})</math>
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\exp(-\ln(1-(1/2)^{1-\frac{1}{\alpha}}))\approx -\ln\varepsilon/\ln 2</math>
:<math>-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)</math>
where ''W'' is the [[Lambert W function]]. So
:<math>(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))</math>
:<math>{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2</math>
:<math>\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)</math>
-->The solution is that ''α'' equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.<ref>{{cite web|title=Global Wealth Report 2013|url=https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|publisher=Credit Suisse|page=22|date=Oct 2013|access-date=2016-01-24|archive-url=https://web.archive.org/web/20150214155424/https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|archive-date=2015-02-14|url-status=dead}}</ref>

The [[Gini coefficient]] is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0,&nbsp;0] and [1,&nbsp;1], which is shown in black (''α''&nbsp;=&nbsp;∞) 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 (for <math>\alpha\ge 1</math>) to be

:<math>G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}</math>

(see Aaberge 2005).

==Random variate generation==
{{further|Non-uniform random variate generation}}

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,&nbsp;1], the variate ''T'' given by

:<math>T=\frac{x_\mathrm{m}}{U^{1/\alpha}}</math>

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=https://books.google.com/books?id=pOGAUcn13fMC|isbn=9780824750886 }}</ref>

==See also==
* {{Annotated link|Bradford's law}}
* {{Annotated link|Gini index}}
* {{Annotated link|Gutenberg–Richter law}}
* {{Annotated link|Matthew effect}}
* {{Annotated link|Pareto analysis}}
* {{Annotated link|Pareto efficiency}}
* {{Annotated link|Pareto interpolation}}
* {{Annotated link|Power law#Power-law probability distributions|Power law probability distributions}}
* {{Annotated link|Sturgeon's law}}
* {{Annotated link|Traffic generation model}}
* {{Annotated link|Zipf's law}}
* {{Annotated link|Heavy-tailed distribution}}

==References==
{{reflist|30em}}

==Notes==
* {{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=209–19 |doi=10.2307/2276207 |bibcode=1905PAmSA...9..209L|jstor=2276207|s2cid=154048722 }}
* {{cite book
| title=Ecrits sur la courbe de la répartition de la richesse
| first=Vilfredo
| last=Pareto
| editor=Librairie Droz
| year=1965
| pages=48
| series=Œuvres complètes : T. III
| isbn=9782600040211}}

* {{cite journal | last = Pareto | first = Vilfredo | year = 1895 | title = La legge della domanda | journal = Giornale degli Economisti | volume = 10 | pages = 59–68 }}
* {{cite book
| first=Vilfredo
| last=Pareto
| year=1896
| title=Cours d'économie politique
}}

==External links==
* {{springer|title=Pareto distribution|id=p/p071580}}
* {{MathWorld |title=Pareto distribution |id=ParetoDistribution}}
* {{citation|mode=cs1
| url=http://www3.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf
| contribution=Gini's Nuclear Family
| first=Rolf
| last=Aabergé
| title=International Conference to Honor Two Eminent Social Scientists
| date=May 2005}}

* {{cite conference
| url=https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
| title=Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes
| first1=Mark E.
| last1=Crovella
| author-link1=Mark Crovella
| first2=Azer
| last2=Bestavros
| conference=IEEE/ACM Transactions on Networking
| volume=5
| number=6
| pages=835–846
| date=December 1997
| access-date=2019-02-25
| archive-date=2016-03-04
| archive-url=https://web.archive.org/web/20160304190612/http://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
| url-status=dead
}}

* [http://www.csee.usf.edu/~kchriste/tools/syntraf1.c syntraf1.c] is a [[C program]] to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.

{{ProbDistributions|continuous-semi-infinite}}

{{Authority control}}

{{DEFAULTSORT:Pareto Distribution}}
[[Category:Actuarial science]]
[[Category:Continuous distributions]]
[[Category:Eponyms in economics]]
[[Category:Power laws]]
[[Category:Probability distributions with non-finite variance]]
[[Category:Management science]]
[[Category:Exponential family distributions]]
[[Category:Vilfredo Pareto]]