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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  $ x_\mathrm{m}>0\, $ scale (real) $ \alpha>0\, $ shape (real) 
Support  $ x \in [x_\mathrm{m}, +\infty)\! $ 
$ \frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m\! $  
cdf  $ 1\left(\frac{x_\mathrm{m}}{x}\right)^\alpha \text{ for } x \ge x_m\! $ 
Mean  $ \frac{\alpha\,x_\mathrm{m}}{\alpha1}\text{ for }\alpha>1\, $ 
Median  $ x_\mathrm{m} \sqrt[\alpha]{2} $ 
Mode  $ x_\mathrm{m}\, $ 
Variance  $ \frac{x_\mathrm{m}^2\alpha}{(\alpha1)^2(\alpha2)}\text{ for }\alpha>2\, $ 
Skewness  $ \frac{2(1+\alpha)}{\alpha3}\,\sqrt{\frac{\alpha2}{\alpha}}\text{ for }\alpha>3\, $ 
Kurtosis  $ \frac{6(\alpha^3+\alpha^26\alpha2)}{\alpha(\alpha3)(\alpha4)}\text{ for }\alpha>4\, $ 
Entropy  $ \ln\left(\frac{x_\mathrm{m}}{\alpha}\right) + \frac{1}{\alpha} + 1\! $ 
mgf  $ \alpha(x_\mathrm{m}t)^\alpha\Gamma(\alpha,x_\mathrm{m}t)\text{ for }t<0\, $ 
Char. func.  $ \alpha(ix_\mathrm{m}t)^\alpha\Gamma(\alpha,ix_\mathrm{m}t)\, $ 
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 Edit
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
 $ \Pr(X>x) = \begin{cases} \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & \text{for }x\ge x_\mathrm{m}, \\ 1 & \text{for } x < x_\mathrm{m}. \end{cases} $
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 Edit
Cumulative distribution functionEdit
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and x_{m} is
 $ F_X(x) = \begin{cases} 1\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & \text{for } x \ge x_\mathrm{m}, \\ 0 & \text{for }x < x_\mathrm{m}. \end{cases} $
Graphical representationEdit
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 Edit
It follows (by differentiation) that the probability density function is
 $ f_X(x)= \begin{cases} \alpha\,\dfrac{x_\mathrm{m}^\alpha}{x^{\alpha+1}} & \text{for }x > x_\mathrm{m}, \\[12pt] 0 & \text{for } x < x_\mathrm{m}. \end{cases} $
Moments and characteristic function Edit
 The expected value of a random variable following a Pareto distribution with α > 1 is

 $ E(X)=\frac{\alpha x_\mathrm{m}}{\alpha1} \, $
 (if α ≤ 1, the expected value does not exist).
 The variance is

 $ \mathrm{Var}(X)=\left(\frac{x_\mathrm{m}}{\alpha1}\right)^2 \frac{\alpha}{\alpha2}. $
 (If α ≤ 2, the variance does not exist.)
 The raw moments are

 $ \mu_n'=\frac{\alpha x_\mathrm{m}^n}{\alphan}, \, $
 but the nth moment exists only for n < α.
 The moment generating function is only defined for nonpositive values t ≤ 0 as

 $ M\left(t,\alpha,x_\mathrm{m}\right) = E(e^{tX}) = \alpha(x_\mathrm{m} t)^\alpha\Gamma(\alpha,x_\mathrm{m} t)\text{ and }M\left(0,\alpha,x_\mathrm{m}\right)=1.\, $
 The characteristic function is given by

 $ \varphi(t;\alpha,x_\mathrm{m})=\alpha(ix_\mathrm{m} t)^\alpha\Gamma(\alpha,ix_\mathrm{m} t), $
 where Γ(a, x) is the incomplete gamma function.
Degenerate case Edit
The Dirac delta function is a limiting case of the Pareto density:
 $ \lim_{\alpha\rightarrow \infty} f(x;\alpha,x_\mathrm{m})=\delta(xx_\mathrm{m}). \, $
Conditional distributions Edit
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 Edit
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.
ApplicationsEdit
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]}
 The standardized price returns on individual stocks ^{[4]}
 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]}
 Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.^{[6]}^{[7]}
Relation to other distributionsEdit
Relation to the exponential distribution Edit
The Pareto distribution is related to the exponential distribution as follows. If X is Paretodistributed with minimum x_{m} and index α, then
 $ Y = \log\left(\frac{X}{x_\mathrm{m}}\right). $
is exponentially distributed with intensity (rate parameter) α. Equivalently, if Y is exponentially distributed with intensity α, then
 $ x_\mathrm{m} e^Y \, $
is Paretodistributed with minimum x_{m} and index α.
This can be shown using the standard change of variable techniques:
 $ \Pr(Y<y) = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) = \Pr(X<x_\mathrm{m} e^y) = 1\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1e^{\alpha y}. $
The last expression is the cumulative distribution function of an exponential distribution with intensity α.
Relation to the lognormal distributionEdit
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 distributionEdit
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 Edit
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" Edit
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^{[8]} 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 Edit
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
 $ L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)} xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'} $
where x(F) is the inverse of the CDF. For the Pareto distribution,
 $ x(F)=\frac{x_\mathrm{m}}{(1F)^{1/\alpha}} $
and the Lorenz curve is calculated to be
 $ L(F) = 1(1F)^{11/\alpha},\, $
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
 $ G = 12\int_0^1L(F)\,dF = \frac{1}{2\alpha1} $ (??  Talk:Pareto distribution#Gini coeff)
(see Aaberge 2005).
Parameter estimation Edit
The likelihood function for the Pareto distribution parameters α and x_{m}, given a sample x = (x_{1}, x_{2}, ..., x_{n}), is
 $ L(\alpha, x_\mathrm{m}) = \prod _{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod _{i=1}^n \frac 1 {x_i^{\alpha+1}}. \! $
Therefore, the logarithmic likelihood function is
 $ \ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m}  (\alpha + 1) \sum _{i=1} ^n \ln x_i. \! $
It can be seen that $ \ell(\alpha, x_\mathrm{m}) $ is monotonically increasing with $ x_\mathrm{m} $, that is, the greater the value of $ x_\mathrm{m} $, the greater the value of the likelihood function. Hence, since $ x \ge x_\mathrm{m} $, we conclude that
 $ \widehat x_\mathrm{m} = \min_i {x_i}. $
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
 $ \frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m}  \sum _{i=1}^n \ln x_i = 0. $
Thus the maximum likelihood estimator for α is:
 $ \widehat \alpha = \frac{n}{\sum _i \left( \ln x_i  \ln \widehat x_\mathrm{m} \right)}. $
The expected statistical error is:
 $ \sigma = \frac {\widehat \alpha} {\sqrt n}. $^{[9]}
Graphical representationEdit
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 Edit
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
 $ T=\frac{x_\mathrm{m}}{U^{1/\alpha}} $
is Paretodistributed.^{[citation needed]} If U is uniformly distributed on [0, 1), it can be exchanged for (1  U).
Bounded Pareto distribution Edit
 See also: Truncated distribution
Probability density function  
Cumulative distribution function  
Parameters  $ L > 0 \, $ location (real) 
Support  $ L \leqslant x \leqslant H \, $ 
$ \frac{\alpha L^\alpha x^{\alpha  1}}{1\left(\frac{L}{H}\right)^\alpha} $  
cdf  $ \frac{1L^\alpha x^{\alpha}}{1\left(\frac{L}{H}\right)^\alpha} $ 
Mean  $ \frac{L^\alpha}{1  \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha1}\right) \cdot \left(\frac{1}{L^{\alpha1}}  \frac{1}{H^{\alpha1}}\right), \alpha\neq 1 $ 
Median  $ L \left(1 \frac{1}{2}\left(1\left(\frac{L}{H}\right)^\alpha\right)\right)^{\frac{1}{\alpha}} $ 
Mode  
Variance  $ \frac{L^\alpha}{1  \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha2}\right) \cdot \left(\frac{1}{L^{\alpha2}}  \frac{1}{H^{\alpha2}}\right), \alpha\neq 2 $ 
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
 $ \frac{\alpha L^\alpha x^{\alpha  1}}{1\left(\frac{L}{H}\right)^\alpha} $
where L ≤ x ≤ H, and α > 0.
Generating bounded Pareto random variables Edit
If U is uniformly distributed on (0, 1), then
 $ \left(\frac{U H^\alpha  U L^\alpha  H^\alpha}{H^\alpha L^\alpha}\right)^{\frac{1}{\alpha}} $
is bounded Paretodistributed.^{[citation needed]}
Symmetric Pareto distributionEdit
The symmetric Pareto distribution can be defined by the probability density function:^{[10]}
 $ f(x;\alpha,x_\mathrm{m}) = \begin{cases} (\alpha x_\mathrm{m}^\alpha/2) x^{\alpha1} & \text{for }x>x_\mathrm{m} \\ 0 & \text{otherwise}. \end{cases} $
It has a similar shape to a Pareto distribution for $ x > x_\mathrm{m} $ while looking like an inverted Pareto distribution for $ x < x_\mathrm{m} $^{[citation needed]}.
See alsoEdit
 Cumulative frequency analysis
 Pareto analysis
 Pareto efficiency
 Pareto interpolation
 Pareto principle
 The Long Tail
 Traffic generation model
NotesEdit
 ↑ 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} 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.
 ↑ 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.
ReferencesEdit
 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 linksEdit
 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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