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第722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>+
→定义
:: <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).
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>。
[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 [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number <math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index <math>\alpha</math> but with minimum <math>x_1</math> instead of <math>x_\text{m}</math>.
The [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number <math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index <math>\alpha</math> but with minimum <math>x_1</math> instead of <math>x_\text{m}</math>.
对于一个符合帕累托分布随机变量的条件概率分布cdf，如果它大于或等于某个特定的数number ，<nowiki><math>x_1</math></nowiki> > <nowiki><math>x_\text{m}</math></nowiki>则称他们具有相同的帕累托指数的帕累托分布index ，但是具有最小的<math>x_1</math>而不是<math>x_\text{m}</math>。
===A characterization theorem一个特征定理===
===A characterization theorem一个特征定理===
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>