# Pareto distribution

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena.

## Definition

If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

${\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The Pareto Type I distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. 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 xm is

$F_{X}(x)={\begin{cases}1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\0&x 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.

### Probability density function

It follows (by differentiation) that the probability density function is

$f_{X}(x)={\begin{cases}{\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}&x\geq x_{\mathrm {m} },\\0&x ### Moments and characteristic function

$E(X)={\begin{cases}\infty &\alpha \leq 1,\\{\frac {\alpha x_{\mathrm {m} }}{\alpha -1}}&\alpha >1.\end{cases}}$ $\mathrm {Var} (X)={\begin{cases}\infty &\alpha \in (1,2],\\\left({\frac {x_{\mathrm {m} }}{\alpha -1}}\right)^{2}{\frac {\alpha }{\alpha -2}}&\alpha >2.\end{cases}}$ (If α ≤ 1, the variance does not exist.)
$\mu _{n}'={\begin{cases}\infty &\alpha \leq n,\\{\frac {\alpha x_{\mathrm {m} }^{n}}{\alpha -n}}&\alpha >n.\end{cases}}$ $M\left(t;\alpha ,x_{\mathrm {m} }\right)=E\left[e^{tX}\right]=\alpha (-x_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-x_{\mathrm {m} }t)$ $M\left(0,\alpha ,x_{\mathrm {m} }\right)=1.$ $\varphi (t;\alpha ,x_{\mathrm {m} })=\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t),$ where Γ(ax) is the incomplete gamma function.

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} ### Geometric mean The geometric mean (G) is $G=x_{\text{m}}\exp \left({\frac {1}{\alpha }}\right)$ . ### Harmonic mean The harmonic mean (H) is $H=x_{\text{m}}\left(1+{\frac {1}{\alpha }}\right)$ . ## Generalized Pareto distributions {{#invoke:see also|seealso}} There is a hierarchy  of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions. Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto 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. In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ. 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 $P(IV)(\sigma ,\sigma ,1,\alpha )=P(I)(\sigma ,\alpha ),$ $P(IV)(\mu ,\sigma ,1,\alpha )=P(II)(\mu ,\sigma ,\alpha ),$ $P(IV)(\mu ,\sigma ,\gamma ,1)=P(III)(\mu ,\sigma ,\gamma ).$ 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. ### 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 $f(y)={\frac {y^{\gamma _{1}-1}(1-y)^{\gamma _{2}-1}}{B(\gamma _{1},\gamma _{2})}},\qquad 00,$ where B( ) is the beta function. If $W=\mu +\sigma (Y^{-1}-1)^{\gamma },\qquad \sigma >0,\gamma >0,$ then W has a Feller–Pareto distribution FP(μ, σ, γ, γ1, γ2). $W=\mu +\sigma \left({\frac {U_{1}}{U_{2}}}\right)^{\gamma }$ and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are $FP(\sigma ,\sigma ,1,1,\alpha )=P(I)(\sigma ,\alpha )$ $FP(\mu ,\sigma ,1,1,\alpha )=P(II)(\mu ,\sigma ,\alpha )$ $FP(\mu ,\sigma ,\gamma ,1,1)=P(III)(\mu ,\sigma ,\gamma )$ $FP(\mu ,\sigma ,\gamma ,1,\alpha )=P(IV)(\mu ,\sigma ,\gamma ,\alpha ).$ ## 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. 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. 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. 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. (Note that the Pareto distribution is not realistic for wealth for the lower end. 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: • 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) • Hard disk drive error rates • Clusters of Bose–Einstein 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){{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=

{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

• The standardized price returns on individual stocks 
• Sizes of sand particles 
• Sizes of meteorites

### 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 "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 α = log4(5) = log(5)/log(4), approximately 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown 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.

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.

## Lorenz curve and Gini coefficient Lorenz curves for a number of Pareto distributions. The case α = ∞ corresponds to perfectly equal distribution (G = 0) and the α = 1 line corresponds to complete inequality (G = 1)

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

$L(F)={\frac {\int _{x_{\mathrm {m} }}^{x(F)}xf(x)\,dx}{\int _{x_{\mathrm {m} }}^{\infty }xf(x)\,dx}}={\frac {\int _{0}^{F}x(F')\,dF'}{\int _{0}^{1}x(F')\,dF'}}$ where x(F) is the inverse of the CDF. For the Pareto distribution,

$x(F)={\frac {x_{\mathrm {m} }}{(1-F)^{\frac {1}{\alpha }}}}$ and the Lorenz curve is calculated to be

$L(F)=1-(1-F)^{1-{\frac {1}{\alpha }}},$ Although the numerator and denominator in the expression for $L(F)$ diverge for $0\leq \alpha <1$ , their ratio does not, yielding L=0 in these cases, which yields a Gini coefficient of unity. 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 (for $\alpha \geq 1$ ) to be

$G=1-2\left(\int _{0}^{1}L(F)dF\right)={\frac {1}{2\alpha -1}}$ (see Aaberge 2005).

## Parameter estimation

The likelihood function for the Pareto distribution parameters α and xm, given a sample x = (x1x2, ..., xn), 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 xm, that is, the greater the value of xm, the greater the value of the likelihood function. Hence, since xxm, 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:

$\sigma ={\frac {\widehat {\alpha }}{\sqrt {n}}}.$ ## 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 xxm,

$\log f_{X}(x)=\log \left(\alpha {\frac {x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}\right)=\log(\alpha x_{\mathrm {m} }^{\alpha })-(\alpha +1)\log x.$ Since α is positive, the gradient −(α+1) is negative.

## Random sample 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

$T={\frac {x_{\mathrm {m} }}{U^{\frac {1}{\alpha }}}}$ is Pareto-distributed. If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).

## Variants

### Bounded Pareto distribution

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 variance in the table on the right should be interpreted as the second moment).

${\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}$ where L ≤ x ≤ H, and α > 0.

#### Generating bounded Pareto random variables

If U is uniformly distributed on (0, 1), then applying inverse-transform method 

$U={\frac {1-L^{\alpha }x^{-\alpha }}{1-({\frac {L}{H}})^{\alpha }}}$ $x=\left(-{\frac {UH^{\alpha }-UL^{\alpha }-H^{\alpha }}{H^{\alpha }L^{\alpha }}}\right)^{-{\frac {1}{\alpha }}}$ is a bounded Pareto-distributed. {{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

### Symmetric Pareto distribution

The symmetric Pareto distribution can be defined by the probability density function:

$f(x;\alpha ,x_{\mathrm {m} })={\begin{cases}{\tfrac {1}{2}}\alpha x_{\mathrm {m} }^{\alpha }|x|^{-\alpha -1}&|x|>x_{\mathrm {m} }\\0&{\text{otherwise}}.\end{cases}}$ It has a similar shape to a Pareto distribution for x > xm and is mirror symmetric about the vertical axis.