Pareto distribution

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. This distribution is also known, mostly outside economics, as the Bradford distribution.

If X is a random variable with a Pareto distribution, then the probability distribution of X is characterized by the statement

${\displaystyle {\rm {P}}(X>x)=\left({\frac {x}{x_{\min }}}\right)^{-k}}$

where x is any number greater than x_min, which is the (necessarily positive) minimum possible value of X, and k is a positive parameter. The family of Pareto distributions is parameterized by two quantities, x_min and k. The density is then

${\displaystyle p(x)=\left\{{\begin{matrix}0,&{\mbox{if }}x<x_{\min };\\\\{k\;x_{\min }^{k} \over x^{k+1}},&{\mbox{if }}x>x_{\min }.\end{matrix}}\right.}$

Pareto distributions are continuous probability distributions. "Zipf's law", also sometimes called the "zeta distribution", may be thought of as a discrete counterpart of the Pareto distribution. The expected value of a random variable following a Pareto distribution is ${\displaystyle x_{\min }\;k \over k-1}$ (if ${\displaystyle k\leq 1}$, the expected value is infinite) and its standard deviation is ${\displaystyle {x_{\min } \over k-1}{\sqrt {k \over k-2}}}$ (if ${\displaystyle k\leq 2}$, the standard deviation doesn't exist).

Examples said to be approximately Pareto distributions:

• wealth distribution in individuals before modern industrial capitalism created the vast middle class
• wealth distribution in individuals even after modern industrial capitalism created the vast middle class
• sizes of human settlements
• visits to Wikipedia pages
• clusters of Bose-Einstein condensate near absolute zero
• file size distribution of Internet traffic which uses the TCP protocol
• value of oil reserves in oil fields
• the number of fatalities due to hurricanes?
• in-degree distribution of blogs in blog link network
• length distribution in jobs assigned supercomputers

See also

• Pareto principle
• Pareto interpolation
• List of measures of inequality
• Gini coefficient

External links

• William J. Reed: The Pareto, Zipf and other power laws, http://linkage.rockefeller.edu/wli/zipf/reed01_el.pdf