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The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. 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, the Pareto distribution has colloquially become known and referred to as the Pareto principle, or "80-20 rule", and is sometimes called the "Matthew principle".<!-- can we find a better reference than https://youtu.be/5WX9UEYZsR8 at 2'10". -->  This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, \alpha (α&nbsp;=&nbsp;log45&nbsp;≈&nbsp;1.16). While \alpha is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena and human activities.

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. 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, the Pareto distribution has colloquially become known and referred to as the Pareto principle, or "80-20 rule", and is sometimes called the "Matthew principle".<!-- can we find a better reference than https://youtu.be/5WX9UEYZsR8 at 2'10". -->  This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, \alpha (α&nbsp;=&nbsp;log45&nbsp;≈&nbsp;1.16). While \alpha is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena and human activities.

以意大利土木工程师、经济学家和社会学家 <font color="#ff8000"> 维尔弗雷多·帕累托Vilfredo Pareto</font>命名的帕累托概率分布，是一种用于描述社会、科学、地球物理、保险精算和许多其他类型的可观测现象的幂律概率。最初用于描述一个社会中的财富分配，符合大部分财富由一小部分人口持有的趋势，<font color="#ff8000"> 帕累托分布</font>被通俗地称为帕雷托法则，或“80-20法则” ，有时也被称为<font color="#ff8000"> “马太原则”</font>。<！——我们能找到比https://youtu.be/5WX9UEYZsR8 at 2'10"更好的参考吗。例如，这条规则规定，一个社会80% 的财富掌握在20% 的人口手中。然而，我们不应该把<font color="#ff8000"> 帕累托分布</font>与帕雷托法则混为一谈，因为前者只对一个特定的幂值产生这个结果，即 α = log45≈1.16。虽然 α 值是一个参数，但经验观察发现80-20分布适用于各种情况，包括自然现象和人类活动。

以意大利土木工程师、经济学家和社会学家 <font color="#ff8000"> 维尔弗雷多·帕累托Vilfredo Pareto</font>命名的帕累托概率分布，是一种用于描述社会、科学、地球物理、保险精算和许多其他类型的可观测现象的幂律概率分布。其最初用于描述一个社会中的财富分配，来符合大部分财富由一小部分人口持有的趋势，<font color="#ff8000"> 帕累托分布</font>被通俗地称为帕雷托法则，或“80-20法则” ，有时也被称为<font color="#ff8000"> “马太原则”</font>。<！——我们能找到比https://youtu.be/5WX9UEYZsR8 at 2'10"更好的参考吗。例如，这条规则规定，一个社会80% 的财富掌握在20% 的人口手中。然而，我们不应该把<font color="#ff8000"> 帕累托分布</font>与帕雷托法则混为一谈，因为前者只对一个特定的幂值产生这个结果，即 α = log45≈1.16。虽然 α 值是一个参数，但经验观察发现80-20分布适用于各种情况，包括自然现象和人类活动。

==Definitions定义==

==Definitions定义==

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= 978-0-89974-012-6|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

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= 978-0-89974-012-6|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

If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

* The [[variance]] of a [[random variable]] following a Pareto distribution is

* The [[variance]] of a [[random variable]] following a Pareto distribution is

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

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

其中 Γ(a,&nbsp;x)是不完全Γ函数。

其中 Γ(a,&nbsp;x)是不完全伽马函数。

The parameters may be solved using the [[method of moments]].<!-- :

The parameters may be solved using the [[method of moments]].

 

系数可能被矩量法来解<!-- :

The parameters may be solved using the method of moments.<!-- :

The parameters may be solved using the method of moments.<!-- :

那么“W”具有Feller-Pareto分布FP(''μ'', ''σ'', ''γ'', ''γ''₁, ''γ''₂)。<ref name=arnold/>

那么“W”具有Feller-Pareto分布FP(''μ'', ''σ'', ''γ'', ''γ''₁, ''γ''₂)。<ref name=arnold/>

  <math>

  <math>

 

《数学》

《数学》

 

 

 

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\begin{align}

 

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

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 |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>

如果<math>U_1 \sim \Gamma(\delta_1, 1)</math> 和 <math>U_2 \sim \Gamma(\delta_2, 1)</math>是相互独立的[[伽马分布|伽马变量]，Feller-Pareto（FP）变量的另一个构造是<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 |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>

如果<math>U_1 \sim \Gamma(\delta_1, 1)</math> 和 <math>U_2 \sim \Gamma(\delta_2, 1)</math>是相互独立的[[伽马分布|伽马变量]，Feller-Pareto（FP）变量的另一个构造是<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 |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}</ref>

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

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

*[[绝对零度]]附近的[[玻色—爱因斯坦凝聚]] 簇<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>

*[[绝对零度]]附近的[[玻色—爱因斯坦凝聚]] 簇<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]] |链接=Special:FilePath/FitParetoDistr.tif]]

[[文件：FitParetoDistr.tif|thumb | 250px |拟合累积帕累托（Lomax）分布到最大一天降雨量，使用[[CumFreq]]，另见[[分布拟合]]]]

[[文件：FitParetoDistr.tif|thumb | 250px |拟合累积帕累托（Lomax）分布到最大一天降雨量，使用[[CumFreq]]，另见[[分布拟合]]]]

|财富分配曲线上的文字

|财富分配曲线上的文字

[[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)]]

[[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)|链接=Special:FilePath/ParetoLorenzSVG.svg]]

[[文件：ParetoLorenzSVG.svg|许多帕累托分布的拇指| 325px |洛伦兹曲线。情形“α”==&nbsp；∞对应于完全相等分布（“G”=&nbsp；0），而“α”==1行对应于完全不等式（“G”=&nbsp；1)]]

[[文件：ParetoLorenzSVG.svg|许多帕累托分布的拇指| 325px |洛伦兹曲线。情形“α”==&nbsp；∞对应于完全相等分布（“G”=&nbsp；0），而“α”==1行对应于完全不等式（“G”=&nbsp；1)]]