Pareto distribution

Helping orphans the way you would do it

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

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.

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 x_min k/(k − 1) (if k = 1, the expected value doesn't exist) and its standard deviation is x_min / (k − 1) √(k/(k − 2)) (for k = 1 or 2 the standard deviation doesn't exist).

Examples of 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
add other examples of Pareto distributions here

If the value of k is chosen judiciously then the Pareto distribution obeys the "80-20 rule.

See also:

External links:

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