Editing Power law (section)

===Lack of well-defined average value===
A power-law <math>x^{-k}</math> has a well-defined [[mean]] over <math>x \in [1,\infty)</math> only if <math> k > 2 </math>, and it has a finite [[variance]] only if <math>k >3</math>; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of [[black swan theory|black swan]] behavior.<ref name=Newman /> This can be seen in the following thought experiment:<ref name="CCSSCS9">{{cite web| url = https://www.youtube.com/watch?v=4uDSEs86xCI| title = 9na CEPAL Charlas Sobre Sistemas Complejos Sociales (CCSSCS): Leyes de potencias| website = [[YouTube]]| date = 31 December 2013}}</ref> imagine a room with your friends and estimate the average monthly income in the room. Now imagine the [[world's richest person]] entering the room, with a monthly income of about 1 [[1,000,000,000|billion]] US$. What happens to the average income in the room? Income is distributed according to a power-law known as the [[Pareto distribution]] (for example, the net worth of Americans is distributed according to a power law with an exponent of 2).

On the one hand, this makes it incorrect to apply traditional statistics that are based on [[variance]] and [[standard deviation]] (such as [[regression analysis]]).<ref>{{Cite journal| doi = 10.1016/j.ijforecast.2020.08.008| issn = 0169-2070| last1 = Taleb| first1 = Nassim Nicholas| last2 = Bar-Yam| first2 = Yaneer| last3 = Cirillo| first3 = Pasquale| title = On single point forecasts for fat-tailed variables| journal = International Journal of Forecasting| date = 2020-10-20| volume = 38| issue = 2| pages = 413–422| pmid = 33100449| pmc = 7572356| s2cid = 220919883| doi-access = free}}</ref> On the other hand, this also allows for cost-efficient interventions.<ref name="CCSSCS9"/> For example, given that car exhaust is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially.<ref>{{cite web |author=Malcolm Gladwell |date=February 13, 2006 |title=Million-Dollar Murray  |url=http://gladwell.com/million-dollar-murray/ |access-date=2015-06-14 |url-status=dead |archive-url=https://web.archive.org/web/20150318142026/http://gladwell.com/million-dollar-murray/ |archive-date=2015-03-18 }}</ref>

The median does exist, however: for a power law ''x''<sup> –''k''</sup>, with exponent {{tmath|k > 1}}, it takes the value 2<sup>1/(''k'' – 1)</sup>''x''<sub>min</sub>, where ''x''<sub>min</sub> is the minimum value for which the power law holds.<ref name= Newman/>