Pareto distribution

The Pareto distribution named after the Italian economist Vilfredo Pareto is a power law distribution found in a large number of real-world situations.

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.

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
• sizes of human settlements
• visits to Encyclopedian pages
• 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:

• 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