Pareto distribution
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Probability density function Pareto probability density functions for various k with x_{m} = 1. The horizontal axis is the x parameter. As k → ∞ the distribution approaches δ(x − x_{m}) where δ is the Dirac delta function. 

Cumulative distribution function Pareto cumulative distribution functions for various k with x_{m} = 1. The horizontal axis is the x parameter. 

Parameters  scale (real) shape (real) 

Support  
Probability density function (pdf)  
Cumulative distribution function (cdf)  
Mean  for k > 1 
Median  
Mode  
Variance  for k > 2 
Skewness  for k > 3 
Excess kurtosis  for k > 4 
Entropy  
Momentgenerating function (mgf)  undefined; see text for raw moments 
Characteristic function 
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution that coincides with social, scientific, geophysical, actuarial, and many other types of observable phenomena. Outside the field of economics it is at times referred to as the Bradford distribution.
Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. This idea is sometimes expressed more simply as the Pareto principle or the "8020 rule" which says that 20% of the population controls 80% of the wealth^{[1]}. It can be seen from the probability density function (PDF) graph on the right, that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. This distribution is not limited to describing wealth or income distribution, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples are sometimes seen as approximately Paretodistributed:
 The sizes of human settlements (few cities, many hamlets/villages)
 File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)
 Clusters of BoseEinstein condensate near absolute zero
 The values of oil reserves in oil fields (a few large fields, many small fields)
 The length distribution in jobs assigned supercomputers (a few large ones, many small ones)
 The standardized price returns on individual stocks
 Sizes of sand particles
 Sizes of meteorites
 Numbers of species per genus (There is subjectivity involved: The tendency to divide a genus into two or more increases with the number of species in it)
 Areas burnt in forest fires
 Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.
[edit] Properties
[edit] Definition
If X is a random variable with a Pareto distribution, then the probability that X is greater than some number x is given by
for all x ≥ x_{m}, where x_{m} 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_{m} and k. When this distribution is used to model the distribution of wealth, then the parameter k is called the Pareto index.
It follows from the above that therefore the cumulative distribution function of a Pareto random variable with parameters k and x_{m} is
[edit] Density function
It follows (by differentiation) that the probability density function is
[edit] Various properties
The expected value of a random variable following a Pareto distribution is
(if k ≤ 1, the expected value is infinite). Its variance is
(If , the variance is infinite). The raw moments are found to be
but they are only defined for k > n. This means that the moment generating function, which is just a Taylor series in x with μ_{n}' / n! as coefficients, is not defined. The characteristic function is given by
where Γ(a,x) is the incomplete gamma function. The Pareto distribution is related to the exponential distribution by
The Dirac delta function is a limiting case of the Pareto distribution:
[edit] A characterization theorem
Suppose X_{i}, i = 1, 2, 3, ... are independent identically distributed random variables whose probability distribution is supported on the interval [k, ∞) for some k > 0. Suppose that for all n, the two random variables min{ X_{1}, ..., X_{n} } and (X_{1} + ... + X_{n})/min{ X_{1}, ..., X_{n} } are independent. Then the common distribution is a Pareto distribution.
[edit] Relation to Zipf's law
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.
[edit] Pareto, Lorenz, and Gini
The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF ƒ or the CDF F as
where x(F) is the inverse of the CDF. For the Pareto distribution,
and the Lorenz curve is calculated to be
where k must be greater than or equal to unity, since the denominator in the expression for L(F) is just the mean value of x. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (k = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated to be
(see Aaberge 2005).
[edit] Parameter estimation
The likelihood function for the Pareto distribution parameters k and x_{m}, given a sample , is
Therefore, the logarithmic likelihood function is
It can be seen that is monotonically increasing with x_{m}, that is, the greater the value of x_{m}, the greater the value of the likelihood function. Hence, since , we conclude that
To find the estimator for k, we compute the corresponding partial derivative and determine where it is zero:
Thus the maximum likelihood estimator for k is:
The expected statistical error is:
[edit] Graphical representation
The characteristic curved 'long tail' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a loglog graph, which then takes the form of a straight line with negative gradient.
[edit] Generating a random sample from Pareto distribution
The Pareto distribution is not yet recognized by many programming languages. In the actuarial field, the Pareto distribution is widely used to estimate portfolio costs. As a matter of fact, it can be quite demanding to get data from this particular probability distribution. One can easily generate a random sample from Pareto distribution by mixing two random variables, which are usually builtin many statistical tools. The process is quite simple; one has to generate numbers from an exponential distribution with its λ equal to a random generated sample from a gamma distribution
and
This process generates data starting at 0, so then we need to add x_{m}.
Alternatively, random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0;1), the variate
is Paretodistributed. [2]
[edit] Bounded Pareto distribution
The bounded Pareto distribution has three parameters and . As in the standard Pareto distribution k determines the shape. L denotes the minimal value, and H denotes the maximal value.
Probability density function 

Cumulative distribution function 

Parameters  location (real) 

Support  
Probability density function (pdf)  
Cumulative distribution function (cdf)  
Mean  
Median  
Mode  
Variance  
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  
Characteristic function 
The probability density function is
where , and k > 0.
[edit] Generating bounded Pareto random variables
If U is uniformly distributed on (0, 1), then
is bounded Paretodistributed [3]
[edit] Generalized Pareto distribution
The family of generalized Pareto distributions (GPD) has three parameters and .
Probability density function 

Cumulative distribution function 

Parameters  location (real) 

Support  
Probability density function (pdf)  where 
Cumulative distribution function (cdf)  
Mean  
Median  
Mode  
Variance  
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  
Characteristic function 
The cumulative distribution function is
for , and when , where is the location parameter, the scale parameter and the shape parameter. Note that some references give the "shape parameter" as .
The probability density function is
again, for , and when .
[edit] Generating generalized Pareto random variables
If U is uniformly distributed on (0, 1], then
This section requires expansion. 
[edit] Annotations
 ^ For a twoquantile population, where 18% of the population owns 82% of the wealth, the Theil index takes the value 1.
[edit] References
 Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association. 9: 209–219.
[edit] See also
[edit] External links
 The Pareto, Zipf and other power laws / William J. Reed  PDF
 Gini's Nuclear Family / Rolf Aabergé.  In: International Conference to Honor Two Eminent Social Scientists, May, 2005  PDF
 The SOCR Resource provides interactive interface to Pareto distribution and Pareto sampling and simulation.
 The CERN cern.jet.random package provides a Java implementation of the Pareto distribution. Downloadable as part of the Scalable Simulation Framework (SSF), among others.