Pareto distribution

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Pareto distribution

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The Pareto distribution is the continuous equivalent to the Zipf distribution, i.e. it is a simple power law function. The distribution arises from Pareto's Law, which states that the number of people in a large population, N, with an income greater than x, can be described by an inverse power law function of the form:

where A and a are parameters to be estimated. When expressed in the form of a probability statement, we have:

which can be interpreted as the probability that a person's income is greater than some amount, x, where k is the minimum income for all people in the population. The cumulative distribution (i.e. the distribution up to an income of x), is therefore:

from which the probability density function can be obtained by differentiation as:

The form of the distribution is shown in the sample plots below. Note that the graph commences at 1, because k=1 and the distribution is only defined for values of x greater than or equal to k. Also note that for the initial value x=k, f(x)=a, as expected in this instance.

Distribution plot, k=1, a=0.5,1,2,3

pareto

Key measures for the Pareto distribution are provided in the table below:

Mean

ak/(a-1), a>1

Mode

k

Variance

ak/[(a-1)2(a-2)], a>2

Skewness

Kurtosis

MGF

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, I. Houghton Mifflin/J Wiley & Sons, New York

Mathworld: Weisstein E W: Pareto distribution: http://mathworld.wolfram.com/ParetoDistribution.html

Wikipedia: Pareto distribution: http://en.wikipedia.org/wiki/Pareto_distribution