Pareto distribution
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Probability density function Pareto probability density functions for various α (labeled "k") with x_{m} = 1. The horizontal axis is the x parameter. As α → ∞ the distribution approaches δ(x − x_{m}) where δ is the Dirac delta function.  
Cumulative distribution function Pareto cumulative distribution functions for various α(labeled "k") with x_{m} = 1. The horizontal axis is the x parameter.  
Parameters  scale (real) shape (real) 
Support  
cdf  
Mean  
Median  
Mode  
Variance  
Skewness  
Kurtosis  
Entropy  
mgf  
Char. func. 
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution that coincides with social, scientific, and many other types of observable phenomena. Outside the field of economics it is sometimes referred to as the Bradford distribution.
Definition
If X is a random variable with a Pareto (Type I) distribution,^{[1]} then the probability that X is greater than some number x is given by
where x_{m} is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The family of Pareto distributions is parameterized by two quantities, x_{m} and α. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
Properties
Cumulative distribution function
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and x_{m} is
Graphical representation
When plotted on linear axes, the distribution assumes the familiar Jshaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are selfsimilar (subject to appropriate scaling factors).
When plotted on logarithmic scales (both axes logarithmic), the distribution is represented by a straight line.
Probability density function
It follows (by differentiation) that the probability density function is
Moments and characteristic function
 The expected value of a random variable following a Pareto distribution with α > 1 is

 (if α ≤ 1, the expected value does not exist).
 The variance is

 (If α ≤ 2, the variance does not exist.)
 The raw moments are

 but the nth moment exists only for n < α.
 The moment generating function is only defined for nonpositive values t ≤ 0 as
 The characteristic function is given by

 where Γ(a, x) is the incomplete gamma function.
Degenerate case
The Dirac delta function is a limiting case of the Pareto density:
Conditional distributions
The conditional probability distribution of a Paretodistributed random variable, given the event that it is greater than or equal to a particular number x_{1} exceeding x_{m}, is a Pareto distribution with the same Pareto index α but with minimum x_{1} instead of x_{m}.
A characterization theorem
Suppose X_{i}, i = 1, 2, 3, ... are independent identically distributed random variables whose probability distribution is supported on the interval [x_{m}, ∞) for some x_{m} > 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.
Applications
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. He also used it to describe distribution of income.^{[2]} 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.^{[3]} However, the 8020 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows 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, 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)^{[4]}
 File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)^{[4]}
 Hard disk drive error rates^{[5]}
 Clusters of Bose–Einstein condensate near absolute zero^{[citation needed]}
 The values of oil reserves in oil fields (a few large fields, many small fields)^{[4]}
 The length distribution in jobs assigned supercomputers (a few large ones, many small ones)^{[citation needed]}
 The standardized price returns on individual stocks ^{[4]}
 Sizes of sand particles ^{[4]}
 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)^{[citation needed]}
 Areas burnt in forest fires
 Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.^{[7]}^{[8]}
 In hydrology the Pareto distribution is applied to extreme events such as annually maximum oneday rainfalls and river discharges. The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum oneday rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
Relation to other distributions
Relation to the exponential distribution
The Pareto distribution is related to the exponential distribution as follows. If X is Paretodistributed with minimum x_{m} and index α, then
is exponentially distributed with intensity (rate parameter) α. Equivalently, if Y is exponentially distributed with intensity α, then
is Paretodistributed with minimum x_{m} and index α.
This can be shown using the standard change of variable techniques:
The last expression is the cumulative distribution function of an exponential distribution with intensity α.
Relation to the lognormal distribution
Note that the Pareto distribution and lognormal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution. (Both of these latter two distributions are "basic" in the sense that the logarithms of their density functions are linear and quadratic, respectively, functions of the observed values.)^{[citation needed]}
Relation to the generalized Pareto distribution
The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.
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.
Relation to the "Pareto principle"
The "8020 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log_{4}5, approximately 1.161. Moreover, the following have been shown^{[9]} to be mathematically equivalent:
 Income is distributed according to a Pareto distribution with index α > 1.
 There is some number 0 ≤ p ≤ 1/2 such that 100p% of all people receive 100(1 − p)% of all income, and similarly for every real (not necessarily integer) n > 0, 100p^{n}% of all people receive 100(1 − p)^{n}% of all income.
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income distribution.
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 α 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 (α = ∞) 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).
Parameter estimation
The likelihood function for the Pareto distribution parameters α and x_{m}, given a sample x = (x_{1}, x_{2}, ..., x_{n}), is
Therefore, the logarithmic likelihood function is
It can be seen that is monotonically increasing with , that is, the greater the value of , the greater the value of the likelihood function. Hence, since , we conclude that
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
Thus the maximum likelihood estimator for α is:
The expected statistical error is:
 ^{[10]}
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.^{[citation needed]}
Generating a random sample from Pareto distribution
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 T given by
is Paretodistributed.^{[citation needed]} If U is uniformly distributed on [0, 1), it can be exchanged for (1  U).
Bounded Pareto distribution
 See also: Truncated distribution
Probability density function  
Cumulative distribution function  
Parameters  location (real) 
Support  
cdf  
Mean  
Median  
Mode  
Variance  
Skewness  
Kurtosis  
Entropy  
mgf  
Char. func. 
The bounded Pareto distribution or truncated Pareto distribution has three parameters α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value. (The Variance in the table on the right should be interpreted as 2nd Moment).
The probability density function is
where L ≤ x ≤ H, and α > 0.
Generating bounded Pareto random variables
If U is uniformly distributed on (0, 1), then
is bounded Paretodistributed.^{[citation needed]}
Symmetric Pareto distribution
The symmetric Pareto distribution can be defined by the probability density function:^{[11]}
It has a similar shape to a Pareto distribution for while looking like an inverted Pareto distribution for ^{[citation needed]}.
See also
 Cumulative frequency analysis
 Pareto analysis
 Pareto efficiency
 Pareto interpolation
 Pareto principle
 The Long Tail
 Traffic generation model
Notes
 ↑ See Arnold (1983).
 ↑ Pareto, Vilfredo, Cours d’Économie Politique: Nouvelle édition par G.H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964, pages 299–345.
 ↑ For a twoquantile population, where approximately 18% of the population owns 82% of the wealth, the Theil index takes the value 1.
 ↑ ^{4.0} ^{4.1} ^{4.2} ^{4.3} ^{4.4} William J. Reed et al., “The Double ParetoLognormal Distribution – A New Parametric Model for Size Distributions”, Communications in Statistics : Theory and Methods 33(8), 17331753, 2004 p 18 et seq.
 ↑ (20100224)Understanding latent sector error and how to protect against them. 8th Usenix Conference on File and Storage Technologies (FAST 2010).
 ↑ Cumfreq, a free computer program for cumulative frequency analysis.
 ↑ Kleiber and Kotz (2003): page 94.
 ↑ (1980). Survival probabilities based on Pareto claim distributions. ASTIN Bulletin 11: 61–71.
 ↑ (2010). Pareto's Law. Mathematical Intelligencer 32 (3): 38–43.
 ↑ M. E. J. Newman (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics 46 (5): 323–351.
 ↑ Grabchak, M. & Samorodnitsky, D.. Do Financial Returns Have Finite or Infinite Variance? A Paradox and an Explanation.
References
 Barry C. Arnold (1983). Pareto Distributions, International Cooperative Publishing House.
 Christian Kleiber and Samuel Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.
 M. O. Lorenz (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association 9 (70): 209–219.
 Pareto V (1965) "La Courbe de la Repartition de la Richesse" (Originally published in 1896). In: Busino G, editor. Oevres Completes de Vilfredo Pareto. Geneva: Librairie Droz. pp. 1–5.
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
 syntraf1.c is a C program to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.
 "SelfSimilarity in World Wide Web Traffic: Evidence and Possible Causes" /Mark E. Crovella and Azer Bestavros
 Eric W. Weisstein, Pareto distribution at MathWorld.
Some common univariate probability distributions  

Continuous  
Discrete  
List of probability distributions 
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