Pareto distribution - Definition and Overview

The Pareto distribution named after the Italian economist Vilfredo Pareto is a power law 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

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

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

<math>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.<math>

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 <math>x_{\min} \; k \over k-1 <math> (if <math> k \leq 1<math>, the expected value is infinite) and its standard deviation is <math>{x_{\min} \over k-1} \sqrt{k \over k-2}<math> (if <math> k \leq 2<math>, 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

External links

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