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. 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 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. 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?

See also

External links

de:Pareto-Verteilung es:Distribución Pareto it:Variabile casuale Paretiana zh:帕累托分布


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