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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 xmin, 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, xmin 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 xmin k/(k-1) (if k=1, the expected value doesn't exist) and its standard deviation is xmin / (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 Wikipedia 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".

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