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→Relation to the "Pareto principle"与“帕累托原则”的关系
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 <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:
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 <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:
“[[80-20定律]”，根据这个定律，20%的人得到所有收入的80%，而20%最富裕的20%的人得到这80%的80%，依此类推，当帕累托指数为<math>\alpha=\log_45=\cfrac{log{10}5}{\log{10}4}\约为1.161</math>时，这条定律成立。这个结果可以从下面给出的[[洛伦兹曲线]]公式中得出。此外，以下内容在数学上是等价的：
“[[80-20定律]]”，根据这个定律，20%的人得到所有收入的80%，而20%最富裕的20%的人得到这80%的80%，依此类推，当帕累托指数为<math>\alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161</math>时，这条定律成立。这个结果可以从下面给出的[[洛伦兹曲线]]公式中得出。此外，以下内容<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>在数学上是等价的：
* Income is distributed according to a Pareto distribution with index ''α'' > 1.
* Income is distributed according to a Pareto distribution with index ''α'' > 1.
*收入按照指数为“α”>1的帕累托分布进行分配。
*收入按照指数为''α'' > 1的帕累托分布进行分配。
* There is some number 0 ≤ ''p'' ≤ 1/2 such that 100''p'' % of all people receive 100(1 − ''p'')% of all income, and similarly for every real (not necessarily integer) ''n'' > 0, 100''p<sup>n</sup>'' % of all people receive 100(1 − ''p'')<sup>''n''</sup> percentage of all income. ''α'' and ''p'' are related by
* There is some number 0 ≤ ''p'' ≤ 1/2 such that 100''p'' % of all people receive 100(1 − ''p'')% of all income, and similarly for every real (not necessarily integer) ''n'' > 0, 100''p<sup>n</sup>'' % of all people receive 100(1 − ''p'')<sup>''n''</sup> percentage of all income. ''α'' and ''p'' are related by
*有一个数字0≤“p”≤“1/2”，即所有人中的100“p”获得全部收入的100%（1–“p”）%，同样，每个实数（不一定是整数）“n”>0，100“p<sup>n</sup>“所有人中有100%（1–“p”）<sup>“n”</sup>“所有人中有100%的人得到100%。”α”和“p”通过以下相关
*有一个数字0 ≤ ''p'' ≤ 1/2，即所有人中的100''p'' %获得全部收入的100(1 − ''p'')%，同样，每个实数（不一定是整数）''n'' > 0，所有人中100''p<sup>n</sup>'' %得到收入的100(1 − ''p'')<sup>''n''</sup> 。”α”和“p”有以下关系：
:: <math>1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)}</math>
:: <math>1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)}</math>