Pareto distribution#Occurrence and applications
{{short description|Probability distribution}}
{{CS1 config|mode=cs1}}
{{Infobox probability distribution
| name =Pareto Type I
| type =density
| pdf_image =File:Probability density function of Pareto distribution.svg
Pareto Type I probability density functions for various with As the distribution approaches where is the Dirac delta function.
| cdf_image =File:Cumulative distribution function of Pareto distribution.svg
Pareto Type I cumulative distribution functions for various with
| parameters = scale (real)
shape (real)
| support =
| pdf =
| cdf =
| quantile =
| mean =
\infty & \text{for }\alpha\le 1 \\
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
\end{cases}
| median =
| mode =
| variance =
\infty & \text{for }\alpha\le 2 \\
\dfrac{x_\mathrm{m}^2\alpha}{(\alpha- 1)^2(\alpha-2)} & \text{for }\alpha>2
\end{cases}
| skewness =
| kurtosis =
| entropy =
| mgf =does not exist
| char =
| fisher =
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
0 & \dfrac{1}{\alpha^2}
\end{bmatrix}
}}
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,{{cite journal |last=Amoroso |first=Luigi|date=January 1938 |title=Vilfredo Pareto |journal=Econometrica (Pre-1986) |via=ProQuest |volume=6 |issue=1 }} is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.{{cite journal |last=Pareto |first=Vilfredo |year=1898 |title=Cours d'economie politique |journal=Journal of Political Economy |volume=6 |doi=10.1086/250536 |url=https://zenodo.org/record/2144014 }}
The Pareto principle or "80:20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value ({{math|α}}) {{nobr|of {{math|log {{sub|4}} 5 ≈ 1.16}}}} precisely reflect it. Empirical observation has shown that this 80:20 distribution fits a wide range of cases, including natural phenomena{{cite journal |last=van Montfort |first=M.A.J. |year=1986 |title=The generalized Pareto distribution applied to rainfall depths |journal=Hydrological Sciences Journal |volume=31 |issue=2 |pages=151–162 |doi=10.1080/02626668609491037 |doi-access=free |bibcode=1986HydSJ..31..151V }} and human activities.{{cite journal |last=Oancea |first=Bogdan |year=2017 |title=Income inequality in Romania: The exponential-Pareto distribution |journal=Physica A: Statistical Mechanics and Its Applications |volume=469 |pages=486–498 |doi=10.1016/j.physa.2016.11.094 |bibcode=2017PhyA..469..486O }}{{cite web |last=Morella |first=Matteo |title=Pareto distribution |url=https://www.academia.edu/59302211 |website=academia.edu}}
Definitions
If X is a random variable with a Pareto (Type I) distribution,{{cite book |first=Barry C. |last=Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn= 978-0-89974-012-6}} then the probability that X is greater than some number x, i.e., the survival function (also called tail function), is given by
:
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
1 & x < x_\mathrm{m},
\end{cases}
where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The type I Pareto distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. If this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
=Cumulative distribution function=
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is
:
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
0 & x < x_\mathrm{m}.
\end{cases}
=Probability density function=
It follows (by differentiation) that the probability density function is
:
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log–log plot, the distribution is represented by a straight line.
Properties
=Moments and characteristic function=
- The expected value of a random variable following a Pareto distribution is
:
::
\frac{\alpha x_{\mathrm{m}}}{\alpha-1} & \alpha>1.
\end{cases}
- The variance of a random variable following a Pareto distribution is
::
\infty & \alpha\in(1,2], \\
\left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha>2.
\end{cases}
: (If α ≤ 2, the variance does not exist.)
- The raw moments are
::
- The moment generating function is only defined for non-positive values t ≤ 0 as
::
::
Thus, since the expectation does not converge on an open interval containing we say that the moment generating function does not exist.
- The characteristic function is given by
::
: where Γ(a, x) is the incomplete gamma function.
The parameters may be solved for using the method of moments.S. Hussain, S.H. Bhatti (2018).
[https://www.researchgate.net/publication/322758024_Parameter_estimation_of_Pareto_distribution_Some_modified_moment_estimators Parameter estimation of Pareto distribution: Some modified moment estimators]. Maejo International Journal of Science and Technology 12(1):11-27.
=Conditional distributions=
The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number exceeding , is a Pareto distribution with the same Pareto index but with minimum instead of :
:
\text{Pr}(X \geq x | X \geq x_1) =
\begin{cases}
\left(\frac{x_1}{x}\right)^\alpha & x \geq x_1, \\
1 & x < x_1.
\end{cases}
This implies that the conditional expected value (if it is finite, i.e. ) is proportional to :
:
In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called the Lindy effect or Lindy's Law.{{cite journal|last1=Eliazar|first1=Iddo|date=November 2017|title=Lindy's Law|journal=Physica A: Statistical Mechanics and Its Applications|volume=486|pages=797–805|bibcode=2017PhyA..486..797E|doi=10.1016/j.physa.2017.05.077|s2cid=125349686 }}
=A characterization theorem=
Suppose are independent identically distributed random variables whose probability distribution is supported on the interval for some . Suppose that for all , the two random variables and are independent. Then the common distribution is a Pareto distribution.{{Citation needed|date=February 2012}}
=Geometric mean=
=Harmonic mean=
The harmonic mean (H) is
:
=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 log-log graph, which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for x ≥ xm,
:
Since α is positive, the gradient −(α + 1) is negative.
Related distributions
=Generalized Pareto distributions=
{{See also|Generalized Pareto distribution}}
There is a hierarchy Johnson, Kotz, and Balakrishnan (1994), (20.4). of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.{{cite book |author1=Christian Kleiber |author2=Samuel Kotz |name-list-style=amp |year=2003 |title=Statistical Size Distributions in Economics and Actuarial Sciences |publisher=Wiley |isbn=978-0-471-15064-0| url=https://books.google.com/books?id=7wLGjyB128IC}} Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto{{cite book|last=Feller |first= W.| year=1971| title=An Introduction to Probability Theory and its Applications| volume=II| edition=2nd | location= New York|publisher=Wiley|page=50}} "The densities (4.3) are sometimes called after the economist Pareto. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ Ax−α as x → ∞". distribution generalizes Pareto Type IV.
==Pareto types I–IV==
The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).
When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.{{cite journal | last1 = Lomax | first1 = K. S. | year = 1954 | title = Business failures. Another example of the analysis of failure data | journal = Journal of the American Statistical Association | volume = 49 | issue = 268| pages = 847–52 | doi=10.1080/01621459.1954.10501239}}
In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.
class="wikitable"
|+Pareto distributions ! !! !! Support !! Parameters | |||
Type I | |||
Type II | |||
Lomax | |||
Type III | |||
Type IV |
The shape parameter α is the tail index, μ is location, σ is scale, γ is an inequality parameter. Some special cases of Pareto Type (IV) are
::
::
::
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.
class="wikitable"
|+Moments of Pareto I–IV distributions (case μ = 0) ! !! !! Condition !! !! Condition | ||||
Type I | ||||
Type II | ||||
Type III | ||||
Type IV |
==Feller–Pareto distribution==
Feller defines a Pareto variable by transformation U = Y−1 − 1 of a beta random variable ,Y, whose probability density function is
:
where B( ) is the beta function. If
:
then W has a Feller–Pareto distribution FP(μ, σ, γ, γ1, γ2).
If and are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is{{cite book |last=Chotikapanich |first=Duangkamon |title=Modeling Income Distributions and Lorenz Curves |chapter=Chapter 7: Pareto and Generalized Pareto Distributions |date=16 September 2008 |pages=121–22 |publisher=Springer |isbn=9780387727967 |chapter-url=https://books.google.com/books?id=fUJZZLj1kbwC}}
:
and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are
:
:
:
:
=Inverse-Pareto Distribution / Power Distribution =
When a random variable follows a pareto distribution, then its inverse follows a Power distribution.
Inverse Pareto distribution is equivalent to a Power distribution Dallas, A. C. "Characterizing the Pareto and power distributions." Annals of the Institute of Statistical Mathematics 28.1 (1976): 491-497.
:
=Relation to the exponential distribution=
The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then
:
is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then
:
is Pareto-distributed with minimum xm and index α.
This can be shown using the standard change-of-variable techniques:
:
\begin{align}
\Pr(Y
& = \Pr(X
\end{align}
The last expression is the cumulative distribution function of an exponential distribution with rate α.
Pareto distribution can be constructed by hierarchical exponential distributions.{{Cite thesis|title=Bayesian semiparametric spatial and joint spatio-temporal modeling|url=https://mospace.umsystem.edu/xmlui/handle/10355/4450|publisher=University of Missouri--Columbia|date=2006|degree=Thesis|first=Gentry|last=White}} section 5.3.1. Let
and
. Then we have and, as a result, .
More in general, if (shape-rate parametrization) and , then .
Equivalently, if and , then .
=Relation to the log-normal distribution=
The Pareto distribution and log-normal 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. (See the previous section.)
=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.
The Pareto distribution with scale and shape is equivalent to the generalized Pareto distribution with location , scale and shape and, conversely, one can get the Pareto distribution from the GPD by taking and if .
=Bounded Pareto distribution=
{{See also|Truncated distribution}}
{{Probability distribution
| name =Bounded Pareto
| type =density
| pdf_image =
| cdf_image =
| parameters =
shape (real)
| support =
| pdf =
| cdf =
| mean =
| median =
| mode =
| variance =
(this is the second raw moment, not the variance)
| skewness =
(this is the kth raw moment, not the skewness)
| kurtosis =
| entropy =
| mgf =
| char =
}}
The bounded (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 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 applying inverse-transform method {{Cite web |url=http://www.cs.bgu.ac.il/~mps042/invtransnote.htm |title=Inverse Transform Method |access-date=2012-08-27 |archive-date=2012-01-17 |archive-url=https://web.archive.org/web/20120117042753/http://www.cs.bgu.ac.il/~mps042/invtransnote.htm |url-status=dead }}
:
:
is a bounded Pareto-distributed.
{{Clear}}
=Symmetric Pareto distribution=
The purpose of the Symmetric and Zero Symmetric Pareto distributions is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from the Pareto distribution. Long probability tails normally means that probability decays slowly, and can be used to fit a variety of datasets. But if the distribution has symmetric structure with two slow decaying tails, Pareto could not do it. Then Symmetric Pareto or Zero Symmetric Pareto distribution is applied instead.{{Cite journal|last=Huang|first=Xiao-dong|date=2004|title=A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic|journal=IEEE Transactions on Broadcasting|volume=50|issue=3|pages=323–334|doi=10.1109/TBC.2004.834013}}
The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:
\tfrac{1}{2}({b \over 2b-X}) ^a & X
1- \tfrac{1}{2}(\tfrac{b}{X})^a& X\geq b
\end{cases}
The corresponding probability density function (PDF) is:
This distribution has two parameters: a and b. It is symmetric about b. Then the mathematic expectation is b. When, it has variance as following:
The CDF of Zero Symmetric Pareto (ZSP) distribution is defined as following:
\tfrac{1}{2}({b \over b-X}) ^a & X<0 \\
1- \tfrac{1}{2}(\tfrac{b}{b+X})^a& X\geq 0
\end{cases}
The corresponding PDF is:
This distribution is symmetric about zero. Parameter a is related to the decay rate of probability and (a/2b) represents peak magnitude of probability.
=Multivariate Pareto distribution=
The univariate Pareto distribution has been extended to a multivariate Pareto distribution.{{cite journal
|last1=Rootzén|first1=Holger |last2=Tajvidi|first2=Nader
|title=Multivariate generalized Pareto distributions |journal=Bernoulli|volume=12|year=2006|number=5 |pages=917–30
|doi=10.3150/bj/1161614952
|citeseerx=10.1.1.145.2991|s2cid=16504396 }}
Statistical inference
=Estimation of parameters=
The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is
:
Therefore, the logarithmic likelihood function is
:
It can be seen that is monotonically increasing with xm, that is, the greater the value of xm, the greater the value of the likelihood function. Hence, since x ≥ xm, 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:{{cite journal |author=M. E. J. Newman |year=2005 |title=Power laws, Pareto distributions and Zipf's law |journal=Contemporary Physics |volume=46 |issue=5 |pages=323–51| arxiv=cond-mat/0412004 |doi=10.1080/00107510500052444 |bibcode=2005ConPh..46..323N|s2cid=202719165 }}
:
Malik (1970){{cite journal |author=H. J. Malik |year=1970 |title=Estimation of the Parameters of the Pareto Distribution |journal=Metrika |volume=15|pages=126–132 |doi=10.1007/BF02613565 |s2cid=124007966 }} gives the exact joint distribution of . In particular, and are independent and is Pareto with scale parameter xm and shape parameter nα, whereas has an inverse-gamma distribution with shape and scale parameters n − 1 and nα, respectively.
Occurrence and applications
=General=
Vilfredo 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.Pareto, Vilfredo, Cours d'Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964, pp. 299–345. [https://web.archive.org/web/20130531151249/http://www.institutcoppet.org/wp-content/uploads/2012/05/Cours-d%C3%A9conomie-politique-Tome-II-Vilfredo-Pareto.pdf Original book archived] This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth.For a two-quantile population, where approximately 18% of the population owns 82% of the wealth, the Theil index takes the value 1. As Michael Hudson points out (The Collapse of Antiquity [2023] p. 85 & n.7) "a mathematical corollary [is] that 10% would have 65% of the wealth, and 5% would have half the national wealth.” However, the 80-20 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.{{citation needed|date=May 2019}} 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. (The Pareto distribution is not realistic for wealth for the lower end, however. In fact, net worth may even be negative.) 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 Pareto-distributed:
File:FitParetoDistr.tif, see also distribution fitting ]]
- The values of oil reserves in oil fields (a few large fields, many small fields)
- The length distribution in jobs assigned to supercomputers (a few large ones, many small ones){{Cite journal|last1=Harchol-Balter|first1=Mor|author1-link=Mor Harchol-Balter|last2=Downey|first2=Allen|date=August 1997|title=Exploiting Process Lifetime Distributions for Dynamic Load Balancing|url=https://users.soe.ucsc.edu/~scott/courses/Fall11/221/Papers/Sync/harcholbalter-tocs97.pdf|journal=ACM Transactions on Computer Systems|volume=15|issue=3|pages=253–258|doi=10.1145/263326.263344|s2cid=52861447}}
- The standardized price returns on individual stocks
- Sizes of sand particles
- The size of meteorites
- Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.Kleiber and Kotz (2003): p. 94.{{cite journal |last1=Seal |first1=H. |year=1980 |title=Survival probabilities based on Pareto claim distributions |journal=ASTIN Bulletin |volume=11 |pages=61–71|doi=10.1017/S0515036100006620 |doi-access=free }}
- Amount of time a user on Steam will spend playing different games. (Some games get played a lot, but most get played almost never.) [https://docs.google.com/spreadsheets/d/1AjtfgTQc1T84NCyJWGcCPN4jrVsOpX0bp0jgPZJEW6A/edit#gid=0]{{Original research inline|date=December 2020}}
- In hydrology the Pareto distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.CumFreq, software for cumulative frequency analysis and probability distribution fitting [https://www.waterlog.info/cumfreq.htm] The blue picture illustrates an example of fitting the Pareto distribution to ranked annually maximum one-day 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.
- In Electric Utility Distribution Reliability (80% of the Customer Minutes Interrupted occur on approximately 20% of the days in a given year).
=Relation to Zipf's law=
The Pareto distribution is a continuous probability distribution. Zipf's law, also sometimes called the zeta distribution, is a discrete distribution, separating the values into a simple ranking. Both are a simple power law with a negative exponent, scaled so that their cumulative distributions equal 1. Zipf's can be derived from the Pareto distribution if the values (incomes) are binned into ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining so that where is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's.
:
where and is an integer representing rank from 1 to N where N is the highest income bracket. So a randomly selected person (or word, website link, or city) from a population (or language, internet, or country) has probability of ranking .
=Relation to the "Pareto principle"=
The "80/20 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 . This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown{{cite journal |last1=Hardy |first1=Michael |year=2010 |title=Pareto's Law |journal=Mathematical Intelligencer |volume=32 |issue=3 |pages=38–43 |doi=10.1007/s00283-010-9159-2|s2cid=121797873 }} 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, 100pn % of all people receive 100(1 − p)n percentage of all income. α and p are related by
::
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 an infinite expected value, and so cannot reasonably model income distribution.
=Relation to Price's law=
Price's law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that . Note that in this case, the total and expected amount of wealth are not defined, and the rule only applies asymptotically to random samples. The extended Pareto Principle mentioned above is a far more general rule.
=Lorenz curve and Gini coefficient=
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 f 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
:
For the denominator is infinite, yielding L=0. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
According to Oxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population.{{cite web|title=62 people own the same as half the world, reveals Oxfam Davos report|url=https://www.oxfam.org/en/pressroom/pressreleases/2016-01-18/62-people-own-same-half-world-reveals-oxfam-davos-report|publisher=Oxfam|date=Jan 2016}} We can estimate the Pareto index that would apply to this situation. Letting ε equal we have:
:
or
:
The solution is that α equals about 1.15, and about 9% of the wealth is owned by each of the two groups. But actually the poorest 69% of the world adult population owns only about 3% of the wealth.{{cite web|title=Global Wealth Report 2013|url=https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|publisher=Credit Suisse|page=22|date=Oct 2013|access-date=2016-01-24|archive-url=https://web.archive.org/web/20150214155424/https://publications.credit-suisse.com/tasks/render/file/?fileID=BCDB1364-A105-0560-1332EC9100FF5C83|archive-date=2015-02-14|url-status=dead}}
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 (for ) to be
:
(see Aaberge 2005).
Random variate generation
{{further|Non-uniform random variate generation}}
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
:
See also
- {{Annotated link|Bradford's law}}
- {{Annotated link|Gutenberg–Richter law}}
- {{Annotated link|Matthew effect}}
- {{Annotated link|Pareto analysis}}
- {{Annotated link|Pareto efficiency}}
- {{Annotated link|Pareto interpolation}}
- {{Annotated link|Power law#Power-law probability distributions|Power law probability distributions}}
- {{Annotated link|Sturgeon's law}}
- {{Annotated link|Traffic generation model}}
- {{Annotated link|Zipf's law}}
- {{Annotated link|Heavy-tailed distribution}}
References
{{reflist|30em}}
Notes
- {{cite journal |author=M. O. Lorenz |year=1905 |title=Methods of measuring the concentration of wealth |journal=Publications of the American Statistical Association |volume=9 |issue=70 |pages=209–19 |doi=10.2307/2276207 |bibcode=1905PAmSA...9..209L|jstor=2276207|s2cid=154048722 }}
- {{cite book
| title=Ecrits sur la courbe de la répartition de la richesse
| first=Vilfredo
| last=Pareto
| editor=Librairie Droz
| year=1965
| pages=48
| series=Œuvres complètes : T. III
| isbn=9782600040211}}
- {{cite journal | last = Pareto | first = Vilfredo | year = 1895 | title = La legge della domanda | journal = Giornale degli Economisti | volume = 10 | pages = 59–68 }}
- {{cite book
| first=Vilfredo
| last=Pareto
| year=1896
| title=Cours d'économie politique
}}
External links
- {{springer|title=Pareto distribution|id=p/p071580}}
- {{MathWorld |title=Pareto distribution |id=ParetoDistribution}}
- {{citation|mode=cs1
| url=http://www3.unisi.it/eventi/GiniLorenz05/25%20may%20paper/PAPER_Aaberge.pdf
| contribution=Gini's Nuclear Family
| first=Rolf
| last=Aabergé
| title=International Conference to Honor Two Eminent Social Scientists
| date=May 2005}}
- {{cite conference
| url=https://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
| title=Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes
| first1=Mark E.
| last1=Crovella
| author-link1=Mark Crovella
| first2=Azer
| last2=Bestavros
| conference=IEEE/ACM Transactions on Networking
| volume=5
| number=6
| pages=835–846
| date=December 1997
| access-date=2019-02-25
| archive-date=2016-03-04
| archive-url=https://web.archive.org/web/20160304190612/http://www.cs.bu.edu/~crovella/paper-archive/self-sim/journal-version.pdf
| url-status=dead
}}
- [http://www.csee.usf.edu/~kchriste/tools/syntraf1.c syntraf1.c] is a C program to generate synthetic packet traffic with bounded Pareto burst size and exponential interburst time.
{{ProbDistributions|continuous-semi-infinite}}
{{Authority control}}
{{DEFAULTSORT:Pareto Distribution}}
Category:Continuous distributions
Category:Probability distributions with non-finite variance