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}}The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,JOURNAL, Amoroso, Luigi, 1938, VILFREDO PARETO, Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest, 6, 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.JOURNAL, Pareto, Vilfredo, 1898, Cours d’economie politique, Journal of Political Economy, 6, 10.1086/250536,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}}) of log45 â‰ˆ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomenaJOURNAL, VAN MONTFORT, M.A.J., 1986, The Generalized Pareto distribution applied to rainfall depths, Hydrological Sciences Journal, 31, 2, 151–162, 10.1080/02626668609491037, free, 1986HydSJ..31..151V, and human activities.JOURNAL, Oancea, Bogdan, 2017, Income inequality in Romania: The exponential-Pareto distribution, Physica A: Statistical Mechanics and Its Applications, 469, 486–498, 10.1016/j.physa.2016.11.094, 2017PhyA..469..486O, JOURNAL, Morella, Matteo, Pareto Distribution,www.academia.edu/59302211, academia.edu,

Definitions

If X is a random variable with a Pareto (Type I) distribution,BOOK, Barry C. Arnold, 1983, Pareto Distributions, International Co-operative Publishing House, 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 & xge 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}, 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 & alphain(1,2], left(frac{x_mathrm{m}}{alpha-1}right)^2 frac{alpha}{alpha-2} & alpha>2.end{cases}

• The raw moments are

• The moment generating function is only defined for non-positive values t â‰¤ 0 as

Thus, since the expectation does not converge on an open interval containing t=0 we say that the moment generating function does not exist.

• The characteristic function is given by

The parameters may be solved for using the method of moments.S. Hussain, S.H. Bhatti (2018). 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 x_1 exceeding x_text{m}, is a Pareto distribution with the same Pareto index alpha but with minimum x_1 instead of x_text{m}. This implies that the conditional expected value (if it is finite, i.e. alpha>1) is proportional to x_1. 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.JOURNAL, Eliazar, Iddo, November 2017, Lindy’s Law, Physica A: Statistical Mechanics and Its Applications, 486, 797–805, 2017PhyA..486..797E, 10.1016/j.physa.2017.05.077, 125349686,

A characterization theorem

Suppose X_1, X_2, X_3, dotsc are independent identically distributed random variables whose probability distribution is supported on the interval [x_text{m},infty) for some x_text{m}>0. Suppose that for all n, the two random variables min{X_1,dotsc,X_n} and (X_1+dotsb+X_n)/min{X_1,dotsc,X_n} are independent. Then the common distribution is a Pareto distribution.{{Citation needed|date=February 2012}}

Geometric mean

The geometric mean (G) isJohnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.

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.BOOK, Christian Kleiber, Samuel Kotz, amp, 2003, Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley & Sons, Wiley, 978-0-471-15064-0,books.google.com/books?id=7wLGjyB128IC, Pareto Type IV contains Pareto Type I–III as special cases. The Feller–ParetoBOOK, Feller, W., 1971, An Introduction to Probability Theory and its Applications, II, 2nd, New York, Wiley, 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 â†’ âˆž”. 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.JOURNAL, Lomax, K. S., 1954, Business failures. Another example of the analysis of failure data, Journal of the American Statistical Association, 49, 268, 847–52, 10.1080/01621459.1954.10501239, In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by Ïƒ.{|class=“wikitable“|+Pareto distributions! !! overline{F}(x)=1-F(x) !! Support !! Parameters| Type I

left[frac x sigma right]^{-alpha}	x ge sigma	sigma > 0, alpha

| Type II

left[1 + frac{x-mu} sigma right]^{-alpha}	x ge mu	mu in mathbb R, sigma > 0, alpha

| Lomax

left[1 + frac x sigma right]^{-alpha}	x ge 0	sigma > 0, alpha

| Type III

left[1 + left(frac{x-mu} sigma right)^{1/gamma}right]^{-1}	x ge mu	mu in mathbb R, sigma, gamma > 0

| Type IV

left[1 + left(frac{x-mu} sigma right)^{1/gamma}right]^{-alpha}	x ge mu	mu in mathbb R, sigma, gamma > 0, alpha

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)! !! operatorname{E}[X] !! Condition !! operatorname{E}[X^delta] !! Condition| Type I< alpha

frac{sigma alpha}{alpha-1}	alpha > 1	frac{sigma^delta alpha}{alpha-delta}	delta

| Type II< delta < alpha

frac{ sigma }{alpha-1}+mu	alpha > 1	frac{ sigma^delta Gamma(alpha-delta)Gamma(1+delta)}{Gamma(alpha)}	0

| Type III

sigmaGamma(1-gamma)Gamma(1 + gamma)	-1