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第1行： − 此词条暂由水流心不竞初译，未经审校，带来阅读不便，请见谅。+ − {{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> − | 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 =<math>\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0</math> − | 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}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\ − -\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2} − \end{bmatrix}</math> − Right: <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> − }} − The '''Pareto distribution''', named after the Italian [[civil engineer]], [[economist]], and sociologist [[Vilfredo Pareto]],<ref>{{Cite journal|last=Amoroso|first=Luigi|date=1938|title=VILFREDO PARETO|url=|journal=Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest|volume=6|pages=|via=}}</ref> 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. 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|date=1898|title=Cours d'economie politique|url=|journal=Journal of Political Economy|volume=6|pages=|via=}}</ref> 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, <math>\alpha</math> (''α'' = log<sub>4</sub>5 ≈ 1.16). While <math>\alpha</math> is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena<ref>{{Cite journal|last=VAN MONTFORT|first=M.A.J.|date=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}}</ref> and human activities.<ref>{{Cite journal|last=Oancea|first=Bogdan|date=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>+ − 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 = log45 ≈ 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== − − 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 − − 如果 X 是一个具有 '''<font color="#ff8000">帕累托分布 （1 型）Pareto (Type i)</font>'''的随机变量，那么 X 大于某个数 x 的概率，即生存函数(也称为'''<font color="#ff8000"> 尾部函数Tail function</font>''')，由以下给出 第81行：
第37行： − </math> + − − − 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]]. − − where xm 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 xm 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. − − 其中，''x''<sub>m</sub>是“X”的最小可能值（必然为正），“α”是一个正参数。<font color="#ff8000">帕累托 i 型分布 Pareto Type I distribution</font>的特征值是[[比例参数]]''x''<sub>m</sub>和[[形状参数]]“α”，即所谓的“尾部指数”。当这个分布被用来模拟财富的分布时，参数“α”被称为[[帕累托指数]]。 第97行：
第46行： − From the definition, the [[cumulative distribution function]] of a Pareto random variable with parameters ''α'' and ''x''<sub>m</sub> is − From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is+ − 根据定义，带有参数 α 和 ''x''<sub>m</sub> 的 Pareto 随机变量的'''<font color="#ff8000"> 累积分布函数Cumulative distribution function </font>'''是 第124行：
第71行： − − \end{cases}</math> − − 结束{ cases } </math > − − It follows (by [[Derivative|differentiation]]) that the [[probability density function]] is − − It follows (by differentiation) that the probability density function is − − 由此可以得出结论: 概率密度函数为 − − − − :<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> − − 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} − − F _ x (x) = begin { cases } frac { alpha x mathrm { m } ^ alpha }{ x ^ alpha + 1} & x ge x mathrm { m } ，0 & x < x mathrm { m }.结束{ cases } − − 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.+ + − 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. − 当在线性轴上绘制时，分布曲线为熟悉的J形曲线，该曲线[[渐近]]地接近每个正交轴。曲线的所有段都是自相似的（取决于适当的比例因子）。在[[双对数图]]中绘制时，分布用直线表示。+ 第157行：
第85行： − + − *服从帕累托分布的[[随机变量] ]的[ [期望值] ]为 − : − 第175行：
第100行： − \end{cases}</math>+ − − 结束{ cases } </math > − − − * 服从帕累托分布的[[随机变量] ]的[ [方差] ]为 − + 第202行：
第122行： − + − − \end{cases}</math> − − 结束{ cases } </math > − − − − : (If ''α'' ≤ 1, the variance does not exist.) − − (If α ≤ 1, the variance does not exist.) − * The raw [[moment (mathematics)|moments]] are+ − − − \mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases} − − N’ = begin { cases } infty & alpha le n，frac { alpha x mathrm { m } ^ n }{ alpha-n } & alpha > n。结束{ cases } − + − − 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) − − 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) − M\left(0,\alpha,x_\mathrm{m}\right)=1.+ − − M 左(0，alpha，x _ mathrm { m }右) = 1。 − − * The [[Characteristic function (probability theory)|characteristic function]] is given by − − \varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),+ − − Varphi (t; alpha，x _ mathrm { m }) = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t) , − − − − : where Γ(''a'', ''x'') is the [[incomplete gamma function]]. − − where Γ(a, x) is the incomplete gamma function. − − 其中 Γ(a, x)是不完全伽马函数。 − + − The parameters may be solved using the [[method of moments]].+ − − 系数可能被矩量法来解<!-- : 第282行：
第166行： − <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> 第929行：
第810行： − + − 《数学》+ − + − + − + − \begin{align}+ − + − 开始{ align }+ − + − 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>+
无编辑摘要
'''帕累托分布 Pareto distribution，'''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<ref>{{Cite journal|last=Amoroso|first=Luigi|date=1938|title=VILFREDO PARETO|url=|journal=Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest|volume=6|pages=|via=}}</ref>，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，拟合大部分财富由一小部分人口持有的分布情况<ref>{{Cite journal|last=Pareto|first=Vilfredo|date=1898|title=Cours d'economie politique|url=|journal=Journal of Political Economy|volume=6|pages=|via=}}</ref>，<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>。例如一个社会80% 的财富掌握在20% 的人口手中这样的社会现象。然而，<font color="#ff8000">'''帕累托分布'''</font>与'''帕雷托法则'''并不能混为一谈，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<ref>{{Cite journal|last=VAN MONTFORT|first=M.A.J.|date=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}}</ref>和人类活动<ref>{{Cite journal|last=Oancea|first=Bogdan|date=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>。
==定义==
如果 X 是一个具有 '''<font color="#ff8000">帕累托分布 （I 型分布）Pareto (Type I)</font>''' 的随机变量<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>，那么 ''X'' 大于某个数 ''x'' 的概率，即'''生存函数 Survival function''' (也称为'''<font color="#ff8000"> 尾部函数Tail function</font>''')，由以下给出：
</math>
</math>
其中，''x<sub>m</sub>''是“''X”''的最小可能值（必为正），“''α''”是一个正参数。<font color="#ff8000">帕累托 I 型分布由</font>[[比例参数]]''x''<sub>m</sub>和[[形状参数]]“α”所决定，即所谓的“'''尾部指数 tail index'''”。当这个分布被用来模拟财富的分布时，参数“''α''”被称为'''[[帕累托指数]]Pareto index'''。
==='''<font color="#ff8000">Cumulative distribution function 累积分布函数</font>'''===
==='''<font color="#ff8000">Cumulative distribution function 累积分布函数</font>'''===
根据定义，带有参数 ''α'' 和 ''x<sub>m</sub>'' 的 Pareto 随机变量的'''<font color="#ff8000"> 累积分布函数Cumulative distribution function </font>'''是
\end{cases}</math>
\end{cases}</math>
===Probability density function概率密度函数===
===Probability density function概率密度函数===
由此可以得出'''概率密度函数Probability density function'''为
:<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><br />
当在线性轴上绘制时，我们默认分布曲线为我们熟知的的J形曲线，该曲线[[渐近]]地接近每个正交轴。曲线的所有段都是自相似的（取决于适当的比例因子）。在[[双对数图|双对数图log-log plot]]中绘制时，分布则是直线表示。
==Properties性质==
==Properties性质==
===Moments and characteristic function矩与特征函数===
===Moments and characteristic function矩与特征函数===
* The [[expected value]] of a [[random variable]] following a Pareto distribution is
*服从帕累托分布的随机变量的'''期望值expected value'''为：
:: <math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
:: <math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
\end{cases}</math>
\end{cases}</math>
* 服从帕累托分布的随机变量的'''方差variance'''为
* The [[variance]] of a [[random variable]] following a Pareto distribution is
:: <math>\operatorname{Var}(X)= \begin{cases}
::<math>\operatorname{Var}(X)= \begin{cases}
<math>\operatorname{Var}(X)= \begin{cases}
<math>\operatorname{Var}(X)= \begin{cases}
左(frac { x _ mathrm { m }{ alpha-1}右) ^ 2 frac { alpha-2} & alpha > 2。
左(frac { x _ mathrm { m }{ alpha-1}右) ^ 2 frac { alpha-2} & alpha > 2。
\end{cases}</math>
\end{cases}</math><br />
(如果 α ≤1，方差不存在.)
(如果 α ≤1，方差不存在.)
*原始的[[矩（数学）|矩]]raw moments为
*原始的[[矩（数学）|矩]]是
:: <math>\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}</math>
:: <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'' ≤ 0 as
* The [[Moment-generating function|moment generating function]] is only defined for non-positive values ''t'' ≤ 0 as
*[[力矩生成函数|力矩生成函数]]仅针对非正值“t”≤0定义为
*仅针对非正''t≤0''的[[力矩生成函数|力矩生成函数]][[Moment-generating function|moment generating function]]定义为
::<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(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>
::<math>M\left(0,\alpha,x_\mathrm{m}\right)=1.</math>
*[[特征函数（概率论）|特征函数]]characteristic function由以下给出
*[[特征函数（概率论）|特征函数]]由以下给出
:: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>
:: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>
其中 Γ(a, x)是'''不完全伽马函数incomplete gamma function'''。
其系数可能被矩量法method of moments来解<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>。<!-- :
The parameters may be solved using the method of moments.<!-- :
The parameters may be solved using the method of moments.<!-- :
Beta = mean * Sqr (mean ^ 2 + var)/(Sqr (var) + Sqr (mean ^ 2 + var)) -- >
Beta = mean * Sqr (mean ^ 2 + var)/(Sqr (var) + Sqr (mean ^ 2 + var)) -- >
那么“W”具有Feller-Pareto分布FP(''μ'', ''σ'', ''γ'', ''γ''<sub>1</sub>, ''γ''<sub>2</sub>)。<ref name=arnold/>
那么“W”具有Feller-Pareto分布FP(''μ'', ''σ'', ''γ'', ''γ''<sub>1</sub>, ''γ''<sub>2</sub>)。<ref name=arnold/>
<math>
<math>
《数学》
\begin{align}
开始{ align }
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>