Editing Pareto distribution (section)

===Relation to the "Pareto principle"===
The "[[Pareto principle|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:
* Income is distributed according to a Pareto distribution with index ''α''&nbsp;>&nbsp;1.
* There is some number 0&nbsp;≤&nbsp;''p''&nbsp;≤&nbsp;1/2 such that 100''p'' % of all people receive 100(1&nbsp;−&nbsp;''p'')% of all income, and similarly for every real (not necessarily integer) ''n''&nbsp;>&nbsp;0, 100''p<sup>n</sup>'' % of all people receive 100(1&nbsp;−&nbsp;''p'')<sup>''n''</sup> percentage of all income. ''α'' and ''p'' are related by
:: <math>1-\frac{1}{\alpha}=\frac{\ln(1-p)}{\ln(p)}=\frac{\ln((1-p)^n)}{\ln(p^n)}</math>

This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.

This excludes Pareto distributions in which&nbsp;0&nbsp;<&nbsp;''α''&nbsp;≤&nbsp;1, which, as noted above, have an infinite expected value, and so cannot reasonably model income distribution.