pareto distribution

\right)^{-k} & \mbox{if }x \ge x_{\min}

               \end{matrix}\right.               |   mean       = $\left \{               \begin{matrix}                 \frac{k x_{min}}{k-1} & \mbox{if }k>1\\                 \infty & \mbox{if }k\le 1               \end{matrix}\right.$ |   median     = $x_{min}2^{1/k}$ |   mode       = $x_{min}$ |   variance   = $\left \{               \begin{matrix}                 \frac{x_{min}^2k}{(k-1)^2(k-2)} & \mbox{if }k>2\\                 \infty & \mbox{if }k\le 2               \end{matrix}\right.$ |   skewness   =|   kurtosis   =|   entropy    =|   mgf        =(undefined)|   char       =

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

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

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 probability density is then

p(x) = \left \{ \begin{matrix} 0, & \mbox{if }x < x_{\min}; \\  \\ k \; x_{\min}^k/x^{k+1}, & \mbox{if }x \ge x_{\min}. \end{matrix} \right.

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  \over k-1

(if k ≤ 1, the expected value is infinite). Its standard deviation is

{x_{\min} \over k-1} \sqrt{k \over k-2}

(if k ≤ 2, the standard deviation is infinite). In non-mathematical terms, this means that there would be a few subjects with many elements or qualities, along with many subjects that each have a few elements or qualities. In economics, the consequence is that most of the purchasing power is held by a few while the rest has limited purchasing power. Overall, the distribution is unequal and the most well known example is expressed by the Pareto Principle where 20% of the population has caused 80% of the results and vice-versa. Examples said to be approximately Pareto distributions:

wealth distribution in individuals
sizes of human settlements
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?
length distribution in jobs assigned supercomputers

External links

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

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Pareto Distribution

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