更改

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

This distribution has two parameters: a and b. It is symmetric by 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>E((x-b)^2)=\int_{-\infty}^{\infty} (x-b)^2p(x)dx={2b^2 \over (a-2)(a-1) }

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

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

 

<math>F(X) = P(x < X ) = \begin{cases}

<math>F(X) = P(x < X ) = \begin{cases}

The corresponding PDF is:

The corresponding PDF is:

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

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

This distribution is symmetric by zero. Parameter is related to the decay rate of probability and represents peak magnitude of probability.<ref name=":0" />

This distribution is symmetric by zero. Parameter is related to the decay rate of probability and represents peak magnitude of probability.<ref name=":0" />

 

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 \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown to be mathematically equivalent:

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 \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown to be mathematically equivalent:

根据“80-20法则” ，20% 的人收入占总收入的80% ，20% 最富裕的20% 收入占总收入的80% ，以此类推，当帕累托指数为 alpha = log 45 = cfrac { log 10}5 log {10}4}大约1.161时，恰好适用。这个结果可以从下面给出的洛伦兹曲线公式推导出来。此外，经证明以下几点在数学上是等价的:

根据“80-20法则” ，20% 的人收入占总收入的80% ，20% 最富裕的20% 收入占总收入的80% ，以此类推，当帕累托指数为 alpha = log 45 = cfrac { log 10}5 log {10}4}大约1.161时，恰好适用。这个结果可以从下面给出的洛伦兹曲线公式推导出来。此外，经证明以下几点在数学上是等价的:

===[[Multivariate Pareto distribution]]===

===[[Multivariate Pareto distribution多元帕累托分布]]===

The univariate Pareto distribution has been extended to a [[multivariate distribution|multivariate]] Pareto distribution.<ref>{{cite journal

The univariate Pareto distribution has been extended to a [[multivariate distribution|multivariate]] Pareto distribution.<ref>{{cite journal

|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader  

|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader  

这排除了0 < α ≤1的帕累托分布，如上所述，这种分布具有无限的期望值，因此不能合理地模拟收入分配。

这排除了0 < α ≤1的帕累托分布，如上所述，这种分布具有无限的期望值，因此不能合理地模拟收入分配。

==Statistical Inference==

==Statistical Inference统计推断==

===Estimation of parameters===

===Estimation of parameters参数估计===

Price's square root law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that \alpha=1. 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.

Price's square root law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that \alpha=1. 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.

The [[likelihood function]] for the Pareto distribution parameters ''α'' and ''x''_m, given an independent [[sample (statistics)|sample]] ''x'' =&nbsp;(''x''₁,&nbsp;''x''₂,&nbsp;...,&nbsp;''x_n''), is

The [[likelihood function]] for the Pareto distribution parameters ''α'' and ''x''_m, given an independent [[sample (statistics)|sample]] ''x'' =&nbsp;(''x''₁,&nbsp;''x''₂,&nbsp;...,&nbsp;''x_n''), 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>

: <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>

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)

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)

一类 Pareto 分布的 Lorenz 曲线。情形 α = ∞对应完全等分布(g = 0) ，α = 1线对应完全不等式(g = 1)

一类Pareto分布的'''<font color="#ff8000"> 洛伦兹曲线Lorenz curve</font>'''。情形α==&nbsp；∞对应于完全相等分布（G&nbsp；=0），α==1线对应于完全不等式（G&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

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

洛伦兹曲线常被用来描述收入和财富的分配。对于任何分布，洛伦兹曲线 l (f)用 PDF 表示，或用 CDF 表示

'''<font color="#ff8000"> 洛伦兹曲线Lorenz curve</font>'''常被用来描述收入和财富的分配。对于任何分布，'''<font color="#ff8000"> 洛伦兹曲线Lorenz curve  L(F)</font>'''用 PDF f  表示，或用 CDF F  表示

It can be seen that <math>\ell(\alpha, x_\mathrm{m})</math> is monotonically increasing with ''x''_m, that is, the greater the value of ''x''_m, the greater the value of the likelihood function. Hence, since ''x'' ≥ ''x''_m, we conclude that

It can be seen that <math>\ell(\alpha, x_\mathrm{m})</math> is monotonically increasing with ''x''_m, that is, the greater the value of ''x''_m, the greater the value of the likelihood function. Hence, since ''x'' ≥ ''x''_m, we conclude that

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

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

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

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

and the Lorenz curve is calculated to be

and the Lorenz curve is calculated to be

For 0<\alpha\le 1 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.

For 0<\alpha\le 1 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.

 

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

: <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>

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>

or

or

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''_m 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.

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''_m 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.

\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})

\ln(1-(1/2)^{1-\frac{1}{\alpha}})=-(\ln\varepsilon/\ln 2)\ln((1/2)^{1-\frac{1}{\alpha}})

- ln (1-(1/2) ^ {1-frac {1}{ alpha }}) exp (- ln (1-(1/2) ^ {1-frac {1}{ alpha }})) approx-ln varepsilon/ln 2

- ln (1-(1/2) ^ {1-frac {1}{ alpha }}) exp (- ln (1-(1/2) ^ {1-frac {1}{ alpha }})) approx-ln varepsilon/ln 2

===General===

===General总则===

-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)

-\ln(1-(1/2)^{1-\frac{1}{\alpha}})\approx W(-\ln\varepsilon/\ln 2)

[[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>Pareto, Vilfredo, ''Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino'', Librairie Droz, Geneva, 1964, pp. 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.{{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:

[[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>Pareto, Vilfredo, ''Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino'', Librairie Droz, Geneva, 1964, pp. 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.{{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:

where W is the Lambert W function. So

where W is the Lambert W function. So

在那里 w 是朗伯W函数。所以

其中 w 是朗伯W函数。所以

<!-- THESE TWO SEEM TO BELONG UNDER [[Zipf's law]] RATHER THAN THE PARETO DISTRIBUTION

<!-- THESE TWO SEEM TO BELONG UNDER [[Zipf's law]] RATHER THAN THE PARETO DISTRIBUTION

(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))

(1/2)^{1-\frac{1}{\alpha}}\approx 1-\exp(-W(-\ln\varepsilon/\ln 2))

* Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)

* Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)

{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2

{1-\frac{1}{\alpha}}\approx -\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2

* Frequencies of [[Given name#Popularity distribution of given names|given names]] -->

* Frequencies of [[Given name#Popularity distribution of given names|given names]] -->

\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)

\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)

* 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|s2cid=154285730 }}</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|s2cid=154285730 }}</ref>

-->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.

-->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.

解决方案是，α 大约等于1.15，这两个群体各拥有大约9% 的财富。但实际上，世界上最贫穷的69% 的成年人只拥有大约3% 的财富。

-->解决方案是，α 大约等于1.15，这两个群体各拥有大约9% 的财富。但实际上，世界上最贫穷的69% 的成年人只拥有大约3% 的财富。

* File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)<ref name ="Reed" />

* 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 |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 (χ² statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.}}</ref>

* [[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 (χ² statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.}}</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 \alpha\ge 1) to be

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 \alpha\ge 1) to be

* 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}}</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}}</ref>

[[File:FitParetoDistr.tif|thumb|250px|Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]

[[File:FitParetoDistr.tif|thumb|250px|Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]] ]]

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

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

* 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 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 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>

(see Aaberge 2005).

(see Aaberge 2005).

* The standardized price returns on individual stocks <ref name="Reed" />

* The standardized price returns on individual stocks <ref name="Reed" />

* Sizes of sand particles <ref name ="Reed" />

* Sizes of sand particles <ref name ="Reed" />

* The size of meteorites

* The size of meteorites

* Male dating success on Tinder [80% of females compete for the 20% most attractive males] <ref name="Medium.com">[https://medium.com/p/2ddf370a6e9a]</ref>

* Male dating success on Tinder [80% of females compete for the 20% most attractive males] <ref name="Medium.com">[https://medium.com/p/2ddf370a6e9a]</ref>

Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0,&nbsp;1], the variate T given by

Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0,&nbsp;1], the variate T given by

* 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>

* 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 (software)|Steam]] will spend playing different games. (Some games get played a lot, but most get played  almost never.) [https://docs.google.com/spreadsheets/d/1BDv2W4IsgxiAhhUznTbMSfRtcLia320Zq1HxzwhKao0/edit#gid=0]

* Amount of time a user on [[Steam (software)|Steam]] will spend playing different games. (Some games get played a lot, but most get played  almost never.) [https://docs.google.com/spreadsheets/d/1BDv2W4IsgxiAhhUznTbMSfRtcLia320Zq1HxzwhKao0/edit#gid=0]

T=\frac{x_\mathrm{m}}{U^{1/\alpha}}

T=\frac{x_\mathrm{m}}{U^{1/\alpha}}

* 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 [[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]].

 

is Pareto-distributed. If U is uniformly distributed on [0,&nbsp;1), it can be exchanged with (1&nbsp;−&nbsp;U).

is Pareto-distributed. If U is uniformly distributed on [0,&nbsp;1), it can be exchanged with (1&nbsp;−&nbsp;U).

是帕累托分布的。如果 u 在[0,1]上是均匀分布的，则它可以与(1-u)交换。

是帕累托分布的。如果U在[0，1）上均匀分布，则可以与（1-U）交换。

===Relation to Zipf's law===

===Relation to Zipf's law===