# Pareto分布

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[1] is a power-law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. 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,[2] the Pareto distribution has colloquially become known and referred to as the Pareto principle, or "80-20 rule", and is sometimes called the "Matthew principle". This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, $\displaystyle{ \alpha }$ (α = log45 ≈ 1.16). While $\displaystyle{ \alpha }$ is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena[3] and human activities.[4]

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. 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, the Pareto distribution has colloquially become known and referred to as the Pareto principle, or "80-20 rule", and is sometimes called the "Matthew principle". This rule states that, for example, 80% of the wealth of a society is held by 20% of its population. However, one should not conflate the Pareto distribution with the Pareto Principle as the former only produces this result for a particular power value, \alpha (α = log45 ≈ 1.16). While \alpha is a parameter, empirical observation has found the 80-20 distribution to fit a wide range of cases, including natural phenomena and human activities.

## Definitions定义

If X is a random variable with a Pareto (Type I) distribution,[5] then the probability that X is greater than some number x, i.e. the survival function (also called tail function), is given by

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

$\displaystyle{ \overline{F}(x) = \Pr(X\gt x) = \begin{cases} \lt math\gt \overline{F}(x) = \Pr(X\gt x) = \begin{cases} \lt math \gt overline { f }(x) = Pr (x \gt x) = begin { cases } \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\ \left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\ 左(frac { x _ mathrm { m }{ x }右) ^ alpha & x ge x _ mathrm { m } , 1 & x \lt x_\mathrm{m}, 1 & x \lt x_\mathrm{m}, 1 & x \lt x mathrm { m } , \end{cases} \end{cases} 结束{ cases } }$

[/itex]

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.

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.

### Cumulative distribution function 累积分布函数

From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is

From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is

$\displaystyle{ F_X(x) = \begin{cases} \lt math\gt F_X(x) = \begin{cases} \lt math \gt f _ x (x) = begin { cases } 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\ 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\ 1-left (frac { x _ mathrm { m }{ x } right) ^ alpha & x ge x _ mathrm { m } , 0 & x \lt x_\mathrm{m}. 0 & x \lt x_\mathrm{m}. 0 & x \lt x mathrm { m }. \end{cases} }$

\end{cases}[/itex]

### Probability density function概率密度函数

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

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

$\displaystyle{ f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x \lt x_\mathrm{m}. \end{cases} }$

f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases}

F _ x (x) = begin { cases } frac { alpha x mathrm { m } ^ alpha }{ x ^ alpha + 1} & x ge x mathrm { m } ，0 & x < x mathrm { m }.结束{ cases }

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.

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性质

### Moments and characteristic function矩与特征函数

$\displaystyle{ \operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\ \lt math\gt \operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\ (x) = begin { cases } infty & alpha le 1, \frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha\gt 1. \frac{\alpha x_\mathrm{m}}{\alpha-1} & \alpha\gt 1. 1.1.1.1.1.2.1.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3. \end{cases} }$

\end{cases}[/itex]

$\displaystyle{ \operatorname{Var}(X)= \begin{cases} \lt math\gt \operatorname{Var}(X)= \begin{cases} { Var }(x) = begin { cases } \infty & \alpha\in(1,2], \\ \infty & \alpha\in(1,2], \\ (1,2)中的 infty & alpha \left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha\gt 2. \left(\frac{x_\mathrm{m}}{\alpha-1}\right)^2 \frac{\alpha}{\alpha-2} & \alpha\gt 2. 左(frac { x _ mathrm { m }{ alpha-1}右) ^ 2 frac { alpha-2} & alpha \gt 2。 \end{cases} }$

\end{cases}[/itex]

(If α ≤ 1, the variance does not exist.)
(If α ≤ 1, the variance does not exist.)


(如果 α ≤1，方差不存在.)

$\displaystyle{ \mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha\gt n. \end{cases} }$
\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}


N’ = begin { cases } infty & alpha le n，frac { alpha x mathrm { m } ^ n }{ alpha-n } & alpha > n。结束{ cases }

$\displaystyle{ M\left(t;\alpha,x_\mathrm{m}\right) = \operatorname{E} \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t) }$

M\left(t;\alpha,x_\mathrm{m}\right) = \operatorname{E} \left [e^{tX} \right ] = \alpha(-x_\mathrm{m} t)^\alpha\Gamma(-\alpha,-x_\mathrm{m} t)

M left (t; alpha，x _ mathrm { m } right) = operatorname { e } left [ e ^ { tX } right ] = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t)

$\displaystyle{ M\left(0,\alpha,x_\mathrm{m}\right)=1. }$

M\left(0,\alpha,x_\mathrm{m}\right)=1.

M 左(0，alpha，x _ mathrm { m }右) = 1。

$\displaystyle{ \varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t), }$
\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),


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.
where Γ(a, x) is the incomplete gamma function.


The parameters may be solved using the method of moments.

beta = mean * Sqr(mean ^ 2 + var) / (Sqr(var) + Sqr(mean ^ 2 + var)) -->

Beta = mean * Sqr (mean ^ 2 + var)/(Sqr (var) + Sqr (mean ^ 2 + var)) -- >

### Conditional distributions条件分布

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}.

The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number $\displaystyle{ x_1 }$ exceeding $\displaystyle{ x_\text{m} }$, is a Pareto distribution with the same Pareto index $\displaystyle{ \alpha }$ but with minimum $\displaystyle{ x_1 }$ instead of $\displaystyle{ x_\text{m} }$.

### A characterization theorem一个特征定理

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.

Suppose $\displaystyle{ X_1, X_2, X_3, \dotsc }$ are independent identically distributed random variables whose probability distribution is supported on the interval $\displaystyle{ [x_\text{m},\infty) }$ for some $\displaystyle{ x_\text{m}\gt 0 }$. Suppose that for all $\displaystyle{ n }$, the two random variables $\displaystyle{ \min\{X_1,\dotsc,X_n\} }$ and $\displaystyle{ (X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\} }$ are independent. Then the common distribution is a Pareto distribution.[citation needed]

### Geometric mean几何平均数

The geometric mean (G) is

The geometric mean (G) is[7] 几何平均值 （“G”）是[7]

 G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).


G = x _ text { m } exp left (frac {1}{ alpha } right).

$\displaystyle{ G = x_\text{m} \exp \left( \frac{1}{\alpha} \right). }$

### Harmonic mean调和平均数

The harmonic mean (H) is 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.

The harmonic mean (H) is[7]

< ! ——在这种情况下，使用 x _ m 作为刻度参数的下限是没有意义的，通常的表示法是 sigma —— >

$\displaystyle{ H = x_\text{m} \left( 1 + \frac{ 1 }{ \alpha } \right). }$

The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).

### Graphical representation图示法

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,

When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.

$\displaystyle{ \log f_X(x)= \log \left(\alpha\frac{x_\mathrm{m}^\alpha}{x^{\alpha+1}}\right) = \log (\alpha x_\mathrm{m}^\alpha) - (\alpha+1) \log x. }$

In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ.

Since α is positive, the gradient −(α + 1) is negative.

{ | class = “ wikitable” border = “1” In this section, the symbol xm, used before to indicate the minimum value of x, is replaced by σ. 在本节中，之前用于表示“x”最小值的符号“x”m将替换为“σ”。
Pareto distributions
+ 帕累托分布

## Related distributions相关分布

\overline{F}(x)=1-F(x) Support Parameters 1-F (x) ! ！支持！参数

### Generalized Pareto distributions广义帕累托分布

Type I 第一类

\left[\frac x \sigma \right]^{-\alpha}

[ frac x sigma right ] ^ {-alpha }

There is a hierarchy [5][8] of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.[5][8][9] Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto[8][10] distribution generalizes Pareto Type IV. 帕累托分布有一个层次结构[5][8]帕累托分布被称为帕累托类型I、II、III、IV，费勒-帕累托分布。[5][8][9]Pareto类型IV包含Pareto类型I-III作为特殊情况。费勒-帕累托[8]引用错误：没有找到与</ref>对应的<ref>标签

\mu \in \mathbb R, \sigma > 0, \alpha

Lomax 洛马克斯

\left[1 + \frac x \sigma \right]^{-\alpha}

[1 + frac x sigma right ] ^ {-alpha }

 $\displaystyle{ \overline{F}(x)=1-F(x) }$ Support Parameters x \ge 0 我们会找到他的，我们会找到他 \sigma > 0, \alpha sigma > 0，alpha Type III 第三类 Type I \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1} 左[1 + 左(frac { x-mu } sigma 右) ^ {1/gamma }右] ^ {-1} $\displaystyle{ \left[\frac x \sigma \right]^{-\alpha} }$ x \ge \mu 我们会找到他的 $\displaystyle{ x \ge \sigma }$ \mu \in \mathbb R, \sigma, \gamma > 0 mu in mathbb r，sigma，gamma > 0 $\displaystyle{ \sigma \gt 0, \alpha }$ Type IV 第 IV 类 Type II \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha} 左[1 + 左(frac { x-mu } sigma 右) ^ {1/gamma }右] ^ {-alpha } $\displaystyle{ \left[1 + \frac{x-\mu} \sigma \right]^{-\alpha} }$ x \ge \mu 我们会找到他的 $\displaystyle{ x \ge \mu }$ \mu \in \mathbb R, \sigma, \gamma > 0, \alpha mu in mathbb r，sigma，gamma > 0，alpha $\displaystyle{ \mu \in \mathbb R, \sigma \gt 0, \alpha }$ Lomax

|| $\displaystyle{ \left[1 + \frac x \sigma \right]^{-\alpha} }$

|| $\displaystyle{ x \ge 0 }$

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

|| $\displaystyle{ \sigma \gt 0, \alpha }$

|-

P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),


P (IV)(sigma，sigma，1，alpha) = p (i)(sigma，alpha) ,

| Type III

P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),

P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),


|| $\displaystyle{ \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-1} }$

P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).


p (IV)(mu，sigma，gamma，1) = p (III)(mu，sigma，gamma).

|| $\displaystyle{ x \ge \mu }$

|| $\displaystyle{ \mu \in \mathbb R, \sigma, \gamma \gt 0 }$

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.

|-

| Type IV

{ | class = “ wikitable” border = “1”
 \operatorname{E}[X] $\displaystyle{ \left[1 + \left(\frac{x-\mu} \sigma \right)^{1/\gamma}\right]^{-\alpha} }$ $\displaystyle{ x \ge \mu }$ $\displaystyle{ \mu \in \mathbb R, \sigma, \gamma \gt 0, \alpha }$ Type I 第一类 \frac{\sigma \alpha}{\alpha-1} frac { sigma alpha }{ alpha-1}

|| \alpha > 1

| | alpha > 1

|| \frac{\sigma^\delta \alpha}{\alpha-\delta}

| | frac { sigma ^ delta alpha }{ alpha-delta }

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 形状参数“α”是尾部索引，“μ”是位置，“σ”是尺度，“γ”是不等式参数。帕累托类型（IV）的一些特殊情况是 || \delta < \alpha

| | delta < alpha

|-

|-

$\displaystyle{ P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha), }$

| Type II

| 第二类

$\displaystyle{ P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha), }$

|| \frac{ \sigma }{\alpha-1}

| | frac { sigma }{ alpha-1}

$\displaystyle{ P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma). }$

|| \alpha > 1

| | alpha > 1

|| \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}

| | frac { sigma ^ delta Gamma (alpha-delta) Gamma (1 + delta)}{ Gamma (alpha)}

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. 均值的有限性、方差的存在性和有限性取决于尾部指数α（不等式指数γ）。特别是，对于某些“δ”>0，分数“δ”矩是有限的，如下表所示，其中“δ”不一定是整数。

|| 0 < \delta < \alpha

| | 0 < delta < alpha

|-

|-

 $\displaystyle{ \operatorname{E}[X] }$ Condition $\displaystyle{ \operatorname{E}[X^\delta] }$ Condition Type III 第三类 \sigma\Gamma(1-\gamma)\Gamma(1 + \gamma) sigma Gamma (1-Gamma) Gamma (1 + Gamma) -1<\gamma<1 -1 < γ < 1 \sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta) sigma ^ delta Gamma (1-Gamma delta) Gamma (1 + Gamma delta) Type I -\gamma^{-1}<\delta<\gamma^{-1} -gamma ^ {-1} < delta < gamma ^ {-1} $\displaystyle{ \frac{\sigma \alpha}{\alpha-1} }$ $\displaystyle{ \alpha \gt 1 }$ Type IV 第 IV 类 $\displaystyle{ \frac{\sigma^\delta \alpha}{\alpha-\delta} }$ \frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)} frac { sigma Gamma (alpha-Gamma) Gamma (1 + Gamma)}{ Gamma (alpha)} $\displaystyle{ \delta \lt \alpha }$ -1<\gamma<\alpha -1 < gamma < alpha \frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)} frac { sigma ^ delta Gamma (alpha-Gamma delta) Gamma (1 + Gamma delta)}{ Gamma (alpha)} Type II -\gamma^{-1}<\delta<\alpha/\gamma -gamma ^ {-1} < delta < alpha/gamma $\displaystyle{ \frac{ \sigma }{\alpha-1} }$ $\displaystyle{ \alpha \gt 1 }$ $\displaystyle{ \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)} }$

|}

|| $\displaystyle{ 0 \lt \delta \lt \alpha }$

|-

| Type III

Feller

|| $\displaystyle{ \sigma\Gamma(1-\gamma)\Gamma(1 + \gamma) }$

|| $\displaystyle{ -1\lt \gamma\lt 1 }$

W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma

W = mu + sigma left (frac { u _ 1}{ u _ 2} right) ^ gamma

|| $\displaystyle{ \sigma^\delta\Gamma(1-\gamma \delta)\Gamma(1+\gamma \delta) }$

|| $\displaystyle{ -\gamma^{-1}\lt \delta\lt \gamma^{-1} }$

and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are

|-

| Type IV

FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha)

FP (sigma，sigma，1,1，alpha) = p (i)(sigma，alpha)

|| $\displaystyle{ \frac{\sigma\Gamma(\alpha-\gamma)\Gamma(1+\gamma)}{\Gamma(\alpha)} }$

FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha)

FP (mu，sigma，1,1，alpha) = p (II)(mu，sigma，alpha)

|| $\displaystyle{ -1\lt \gamma\lt \alpha }$

FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma)

FP (mu，sigma，gamma，1,1) = p (III)(mu，sigma，gamma)

|| $\displaystyle{ \frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)} }$

FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha).

FP (mu，sigma，gamma，1，alpha) = p (IV)(mu，sigma，gamma，alpha).

|| $\displaystyle{ -\gamma^{-1}\lt \delta\lt \alpha/\gamma }$

|-

|-

The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then

|}

 Y = \log\left(\frac{X}{x_\mathrm{m}}\right)


Y = log left (frac { x }{ x _ mathrm { m } right)

#### Feller–Pareto distribution费勒-帕累托分布

Feller[8][10] defines a Pareto variable by transformation U = Y−1 − 1 of a beta random variable Y, whose probability density function is

is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then

$\displaystyle{ f(y) = \frac{y^{\gamma_1-1} (1-y)^{\gamma_2-1}}{B(\gamma_1, \gamma_2)}, \qquad 0\lt y\lt 1; \gamma_1,\gamma_2\gt 0, }$
 x_\mathrm{m} e^Y


2. x mathrm { m } e ^ y

where B( ) is the beta function. If

is Pareto-distributed with minimum xm and index α.

$\displaystyle{ W = \mu + \sigma(Y^{-1}-1)^\gamma, \qquad \sigma\gt 0, \gamma\gt 0, }$

This can be shown using the standard change-of-variable techniques:

then W has a Feller–Pareto distribution FP(μ, σ, γ, γ1, γ2).[5]

\displaystyle{ 《数学》 \begin{align} 开始{ align } If \lt math\gt U_1 \sim \Gamma(\delta_1, 1) } and $\displaystyle{ U_2 \sim \Gamma(\delta_2, 1) }$ are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is[13]


\Pr(Y<y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)<y\right) \\

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=1-e^{-\alpha y}.

1-left (frac { x _ mathrm { m } e ^ y }{ x _ mathrm { m } e ^ y } right) ^ alpha = 1-e ^ {-alpha y }.

$\displaystyle{ W = \mu + \sigma \left(\frac{U_1}{U_2}\right)^\gamma }$

\end{align}

[/itex]

and we write W ~ FP(μ, σ, γ, δ1, δ2). Special cases of the Feller–Pareto distribution are

The last expression is the cumulative distribution function of an exponential distribution with rate α.

$\displaystyle{ FP(\sigma, \sigma, 1, 1, \alpha) = P(I)(\sigma, \alpha) }$
$\displaystyle{ FP(\mu, \sigma, 1, 1, \alpha) = P(II)(\mu, \sigma, \alpha) }$
$\displaystyle{ FP(\mu, \sigma, \gamma, 1, 1) = P(III)(\mu, \sigma, \gamma) }$

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.)

$\displaystyle{ FP(\mu, \sigma, \gamma, 1, \alpha) = P(IV)(\mu, \sigma, \gamma, \alpha). }$

### Relation to the exponential distribution与指数分布的关系

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 is related to the exponential distribution as follows. If X is Pareto-distributed with minimum xm and index α, then

The Pareto distribution with scale x_m and shape \alpha is equivalent to the generalized Pareto distribution with location \mu=x_m, scale \sigma=x_m/\alpha and shape \xi=1/\alpha. Vice versa one can get the Pareto distribution from the GPD by x_m = \sigma/\xi and \alpha=1/\xi.

$\displaystyle{ Y = \log\left(\frac{X}{x_\mathrm{m}}\right) }$

is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then

{{Probability distribution

{概率分布

$\displaystyle{ x_\mathrm{m} e^Y }$
| name       =Bounded Pareto


| 名称=有界帕累托

| type       =density


is Pareto-distributed with minimum xm and index α. 帕累托分布的最小值是“x”m和索引“α”。

| pdf_image  =


| cdf_image  =


This can be shown using the standard change-of-variable techniques: 这可以用变量技术的标准变化来表示：

| parameters =


L > 0 location (real)

L > 0 location (real) < br/>

\displaystyle{ H \gt L location (real)\lt br /\gt H \gt l 位置(real) \lt br/\gt \begin{align} \alpha \gt 0 shape (real) α \gt 0形状(实数) \Pr(Y\lt y) & = \Pr\left(\log\left(\frac{X}{x_\mathrm{m}}\right)\lt y\right) \\ | support =L \leqslant x \leqslant H | 支持= L \莱克斯兰 x \莱克斯兰 H & = \Pr(X\lt x_\mathrm{m} e^y) = 1-\left(\frac{x_\mathrm{m}}{x_\mathrm{m}e^y}\right)^\alpha=1-e^{-\alpha y}. | pdf =\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha} 1-left (frac { l }{ h } right) ^ alpha } \end{align} | cdf =\frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha} | cdf = frac {1-L ^ alpha x ^ {-alpha }{1-left (frac { l }{ h } right) ^ alpha } }
| mean       =


|中值 =

\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1

The last expression is the cumulative distribution function of an exponential distribution with rate α.

\frac\ln\frac{H}{L}, \alpha=1

1.1.1

| median     = L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}}


| median = l left (1-frac {1}{2} left (1-left (1-left (frac { l }{ h } right) ^ alpha right)) ^ {-frac {1}{ alpha }}

### Relation to the log-normal distribution与对数正态分布的关系

| mode       =


2012年10月22日

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.)

| variance   =


| 方差 =

\frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2

### Relation to the generalized Pareto distribution与广义帕累托分布的关系

\frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2

2{ h } ^ 2{ l } ^ 2}{ h } ^ 2-{ l } ^ 2} ln frac { h }{ l } ，alpha = 2

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.

(this is the second raw moment, not the variance)

(这是第二个原始时刻，不是方差)

| skewness   = \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j


| skewness = frac { l ^ { alpha }{1-left (frac { l }{ h } right) ^ { alpha } cdot frac { alpha * (l ^ { k-alpha }-h ^ { k-alpha })}{(alpha-k)}} ，alpha neq j

The Pareto distribution with scale $\displaystyle{ x_m }$ and shape $\displaystyle{ \alpha }$ is equivalent to the generalized Pareto distribution with location $\displaystyle{ \mu=x_m }$, scale $\displaystyle{ \sigma=x_m/\alpha }$ and shape $\displaystyle{ \xi=1/\alpha }$. Vice versa one can get the Pareto distribution from the GPD by $\displaystyle{ x_m = \sigma/\xi }$ and $\displaystyle{ \alpha=1/\xi }$.

(this is the kth raw moment, not the skewness)

（这是第k个原始时刻，不是偏斜）

| kurtosis   =


### Bounded Pareto distribution有界帕累托分布

| entropy    =


| 熵 =

| mgf        =


2012年10月22日

{{Probability distribution 模板:概率分布

}}

| type       =density


| pdf_image  =


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.

| cdf_image  =

| parameters =


$\displaystyle{ L \gt 0 }$ location (real)

$\displaystyle{ H \gt L }$ location (real)

\frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha},


1-left (frac { l }{ h } right) ^ alpha } ,

$\displaystyle{ \alpha \gt 0 }$ shape (real)

| support    =$\displaystyle{ L \leqslant x \leqslant H }$


where L ≤ x ≤ H, and α > 0.

where L ≤ x ≤ H, and α > 0.

| pdf        =$\displaystyle{ \frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha} }$

| cdf        =$\displaystyle{ \frac{1-L^\alpha x^{-\alpha}}{1-\left(\frac{L}{H}\right)^\alpha} }$

| mean       =


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

$\displaystyle{ \frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-1}\right) \cdot \left(\frac{1}{L^{\alpha-1}} - \frac{1}{H^{\alpha-1}}\right), \alpha\neq 1 }$

$\displaystyle{ \frac{{H}{L}}{{H}-{L}}\ln\frac{H}{L}, \alpha=1 }$

U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha}

{1-l ^ alpha x ^ {-alpha }{1-(frac { l }{ h }) ^ alpha }

| median     =$\displaystyle{ L \left(1- \frac{1}{2}\left(1-\left(\frac{L}{H}\right)^\alpha\right)\right)^{-\frac{1}{\alpha}} }$


x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}}

X = left (- frac { u h ^ alpha-u l ^ alpha-h ^ alpha }{ h ^ alpha ^ alpha } right) ^ {-frac {1}{ alpha }}

| mode       =

| variance   =


is a bounded Pareto-distributed.

$\displaystyle{ \frac{L^\alpha}{1 - \left(\frac{L}{H}\right)^\alpha} \cdot \left(\frac{\alpha}{\alpha-2}\right) \cdot \left(\frac{1}{L^{\alpha-2}} - \frac{1}{H^{\alpha-2}}\right), \alpha\neq 2 }$

$\displaystyle{ \frac{2{H}^2{L}^2}{{H}^2-{L}^2}\ln\frac{H}{L}, \alpha=2 }$

(this is the second raw moment, not the variance) （这是第二个原始时刻，不是方差）

| skewness   = $\displaystyle{ \frac{L^{\alpha}}{1-\left(\frac{L}{H}\right)^{\alpha}} \cdot \frac{\alpha * (L^{k-\alpha}-H^{k-\alpha})}{(\alpha-k)}, \alpha \neq j }$


The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. 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.

(this is the kth raw moment, not the skewness) （这是第k个原始时刻，不是偏斜）

| kurtosis   =


The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:

| entropy    =

| mgf        =

| char       =


}}

The likelihood function for the Pareto distribution parameters α and xm, given an independent sample x = (x1, x2, ..., xn), is

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.

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}}.


L (alpha，x mathrm { m }) = prod { i = 1} ^ n alpha frac { x mathrm { m } ^ alpha } = alpha ^ n ^ x mathrm { m }{ n ^ { n alpha } prod { i = 1} ^ n frac {1}{ x i ^ { alpha + 1}}.

Therefore, the logarithmic likelihood function is

$\displaystyle{ \frac{\alpha L^\alpha x^{-\alpha - 1}}{1-\left(\frac{L}{H}\right)^\alpha} }$,

\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum_{i=1} ^n \ln x_i.


Ell (alpha，x _ mathrm { m }) = n ln alpha + n alpha ln x _ mathrm { m }-(alpha + 1) sum { i = 1} ^ n ln x _ i.

where L ≤ x ≤ H, and α > 0.

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 x ≥ xm, we conclude that

#### Generating bounded Pareto random variables生成有界Pareto随机变量

If U is uniformly distributed on (0, 1), then applying inverse-transform method [15]

\widehat x_\mathrm{m} = \min_i {x_i}.


$\displaystyle{ U = \frac{1 - L^\alpha x^{-\alpha}}{1 - (\frac{L}{H})^\alpha} }$

To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:

$\displaystyle{ x = \left(-\frac{U H^\alpha - U L^\alpha - H^\alpha}{H^\alpha L^\alpha}\right)^{-\frac{1}{\alpha}} }$

\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.


{ partial alpha } = frac { n }{ alpha } + n ln x _ mathrm { m }-sum _ { i = 1} ^ n ln x _ i = 0.

is a bounded Pareto-distributed.[citation needed] 是一个有界的帕累托分布。[citation needed]

Thus the maximum likelihood estimator for α is:

### Symmetric Pareto distribution对称帕累托分布

\widehat \alpha = \frac{n}{\sum _i  \ln (x_i/\widehat x_\mathrm{m}) }.


The purpose of Symmetric Pareto distribution and Zero Symmetric Pareto distribution is to capture some special statistical distribution with a sharp probability peak and symmetric long probability tails. These two distributions are derived from Pareto distribution. Long probability tail normally means that probability decays slowly. Pareto distribution performs fitting job in many cases. 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.[17]

The expected statistical error is:

The Cumulative distribution function (CDF) of Symmetric Pareto distribution is defined as following:[17] 对称帕累托分布的累积分布函数（CDF）定义如下：[17]

\sigma = \frac {\widehat \alpha} {\sqrt n}.


1.1.2.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.

$\displaystyle{ F(X) = P(x \lt X ) = \begin{cases} \tfrac{1}{2}({b \over 2b-X}) ^a & X\lt b \\ Malik (1970) gives the exact joint distribution of (\hat{x}_\mathrm{m},\hat\alpha). In particular, \hat{x}_\mathrm{m} and \hat\alpha are independent and \hat{x}_\mathrm{m} is Pareto with scale parameter xm and shape parameter nα, whereas \hat\alpha has an inverse-gamma distribution with shape and scale parameters n − 1 and nα, respectively. Malik (1970)给出了(hat { x } _ mathrm { m } ，hat alpha)的精确联合分布。特别地，hat { x } _ mathrm { m }和 hat alpha 是独立的，hat { x } _ mathrm { m }是带有尺度参数 xm 和形状参数 nα 的 Pareto 分布，而 hat alpha 是带有形状参数 n-1和尺度参数 nα 的反 γ 分布。 1- \tfrac{1}{2}(\tfrac{b}{X})^a& X\geq b \end{cases} }$

The corresponding probability density function (PDF) is:[17]

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. 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. (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:

Vilfredo Pareto最初用这种分布来描述个人之间的财富分配，因为它似乎很好地表明，任何社会的大部分财富都是由社会中较小比例的人拥有的。他还用它来描述收入分配。这种观点有时更简单地表述为帕累托原则或“80-20法则”，即20%的人口控制着80%的财富。然而，80-20规则对应于一个特定的α值，事实上，帕累托在其经济政治课程中关于英国所得税的数据表明，大约30%的人口拥有约70%的收入。本文开头的概率密度函数（PDF）图显示，人均拥有少量财富的人口的“概率”或比例相当高，然后随着财富的增加而稳步下降。（然而，对于低端财富而言，帕累托分布并不现实。事实上，净资产甚至可能是负的。）这种分布不仅限于描述财富或收入，而且在许多情况下，在从“小”到“大”的分配中能够找到平衡。以下示例有时被视为近似帕累托分布：

\alpha\approx 1/(1+\ln(1-\exp(-W(-\ln\varepsilon/\ln 2)))/\ln 2)

(1-exp (- w (- ln varepsilon/ln 2))/ln 2)

• The sizes of human settlements (few cities, many hamlets/villages)[18][19]
• 人类居住区的规模（少数城市，许多村庄/村庄）[18][19]

-->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.

-->解决方案是，α 大约等于1.15，这两个群体各拥有大约9% 的财富。但实际上，世界上最贫穷的69% 的成年人只拥有大约3% 的财富。

• File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)[18]
• 使用TCP协议的Internet流量的文件大小分布（许多较小的文件，少数较大的文件）[20]

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\ge 1) to be

Fitted cumulative Pareto (Lomax) distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting

[[文件：FitParetoDistr.tif|thumb | 250px |拟合累积帕累托（Lomax）分布到最大一天降雨量，使用CumFreq，另见分布拟合]]

G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1}

1-2 left (int _ 0 ^ 1L (f) ，dF right) = frac {1}{2 alpha-1}

• The length distribution in jobs assigned to supercomputers (a few large ones, many small ones)[24]
• 分配给超级计算机的作业长度分布（一些大型计算机，许多小型计算机）[25]

(see Aaberge 2005).

(见 Aaberge 2005)。

• The standardized price returns on individual stocks [18]
• 个股的标准化价格回报率[20]
• Sizes of sand particles [18]
• The size of meteorites
• 陨石的大小
• Male dating success on Tinder [80% of females compete for the 20% most attractive males] [26]
• 在Tinder上，男性约会成功【80%的女性竞争20%最具吸引力的男性】[26]

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

• Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.[27][28]
• 某些业务领域（如一般责任险、商用车等）的大额伤亡损失的严重程度，[29][30]
• Amount of time a user on Steam will spend playing different games. (Some games get played a lot, but most get played almost never.) [1]
• 用户在 Steam上玩不同游戏的时间。（有些游戏经常玩，但大多数几乎从不玩。）[2]

T=\frac{x_\mathrm{m}}{U^{1/\alpha}}

1/alpha }}

• 水文学中，帕累托分布适用于极端事件，例如对于一天内的最大降雨事件，应用软件进行频率分布拟合[32]蓝色图片说明了一个拟合帕累托分布的例子根据二项分布对年最大单日降雨量进行排名，也显示了90%的置信带。降雨数据由绘图位置s表示，作为累积频率分析的一部分。

is Pareto-distributed. If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).

### Relation to Zipf's law与齐普夫定律的关系

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 $\displaystyle{ x }$ values (incomes) are binned into $\displaystyle{ N }$ ranks so that the number of people in each bin follows a 1/rank pattern. The distribution is normalized by defining $\displaystyle{ x_m }$ so that $\displaystyle{ \alpha x_\mathrm{m}^\alpha = \frac{1}{H(N,\alpha-1)} }$ where $\displaystyle{ H(N,\alpha-1) }$ is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's.

$\displaystyle{ f(x) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} = \frac{1}{x^s H(N,s)} }$

where $\displaystyle{ s = \alpha-1 }$ and $\displaystyle{ x }$ 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 $\displaystyle{ f(x) }$ probability of ranking $\displaystyle{ x }$.

### Relation to the "Pareto principle"与“帕累托原则”的关系

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 $\displaystyle{ \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161 }$. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown[33] to be mathematically equivalent:

80-20定律”，根据这个定律，20%的人得到所有收入的80%，而20%最富裕的20%的人得到这80%的80%，依此类推，当帕累托指数为$\displaystyle{ \alpha = \log_4 5 = \cfrac{\log_{10} 5}{\log_{10} 4} \approx 1.161 }$时，这条定律成立。这个结果可以从下面给出的洛伦兹曲线公式中得出。此外，以下内容[34]在数学上是等价的：

• Income is distributed according to a Pareto distribution with index α > 1.
• 收入按照指数为α > 1的帕累托分布进行分配。
• 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
• 有一个数字0 ≤ p ≤ 1/2，即所有人中的100p %获得全部收入的100(1 − p)%，同样，每个实数（不一定是整数）n > 0，所有人中100pn %得到收入的100(1 − p)n 。”α”和“p”有以下关系：
$\displaystyle{ 1-\frac{1}{\alpha}=\frac{\ln(1-p^n)}{\ln(1-(1-p)^n)} }$

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.

### Relation to Price's law与价格定律的关系

Price's square root law is sometimes offered as a property of or as similar to the Pareto distribution. However, the law only holds in the case that $\displaystyle{ \alpha=1 }$. 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.

Price的平方根定律有时作为帕累托分布的属性或类似于帕累托分布提供。然而，该定律只适用于$\displaystyle{ \alpha=1 }$的情况。请注意，在这种情况下，没有定义财富的总量和预期金额，而且该规则只适用于渐近随机样本。上面提到的扩展帕累托原则是一个更一般的规则。

### Lorenz curve and Gini coefficient洛伦兹曲线与基尼系数

| title=Ecrits sur la courbe de la répartition de la richesse

|关于财富分配曲线的文章

| title=Ecrits sur la courbe de la répartition de la richesse

|财富分配曲线上的文字

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)

| first=Vilfredo

| last=Pareto

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 Lorenz曲线通常用于描述收入和财富分配。对于任何分布，洛伦兹曲线“L”（“F”）用PDF“F”或CDF“F”表示为

| editor=Librairie Droz

| year=1965

1965年

$\displaystyle{ L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'} }$

| pages=48

48

| series=Œuvres complètes : T. III

| series=Œuvres complètes : T. III

where x(F) is the inverse of the CDF. For the Pareto distribution,

| isbn=9782600040211}}

9782600040211}

$\displaystyle{ x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}} }$

and the Lorenz curve is calculated to be

| first=Vilfredo

| last=Pareto

$\displaystyle{ L(F) = 1-(1-F)^{1-\frac{1}{\alpha}}, }$

| year=1896

1896年

| title=Cours d'économie politique

| title=Cours d'économie politique

For $\displaystyle{ 0\lt \alpha\le 1 }$ 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. 对于$\displaystyle{ 0\lt \alpha\le 1 }$分母是无穷大的，得到“L”=0。右图显示了一些Pareto分布的Lorenz曲线示例。

| doi=10.1177/000271629700900314| s2cid=143528002

10.1177/000271629700900314 | s2cid = 143528002

}}

}}

According to Oxfam (2016) the richest 62 people have as much wealth as the poorest half of the world's population.[35] We can estimate the Pareto index that would apply to this situation. Letting ε equal $\displaystyle{ 62/(7\times 10^9) }$ we have:

$\displaystyle{ L(1/2)=1-L(1-\varepsilon) }$

or

$\displaystyle{ 1-(1/2)^{1-\frac{1}{\alpha}}=\varepsilon^{1-\frac{1}{\alpha}} }$

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.[37]

| title=Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes

| title = 万维网流量的自相似性: 证据和可能的原因

| first1=Mark E.

1 = Mark e.

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 $\displaystyle{ \alpha\ge 1 }$) to be

| last1=Crovella

1 = Crovella

| author-link1=Mark Crovella

1 = Mark Crovella

$\displaystyle{ G = 1-2 \left (\int_0^1L(F) \, dF \right ) = \frac{1}{2\alpha-1} }$

| first2=Azer

2 = Azer

| last2=Bestavros

2 = Bestavros

(see Aaberge 2005).

| conference=IEEE/ACM Transactions on Networking

| volume=5

5

## Computational methods计算方法

| number=6

6

### Random sample generation随机样本生成

| pages=835–846

| 页数 = 835-846

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

| date=December 1997}}

| date = December 1997}

$\displaystyle{ T=\frac{x_\mathrm{m}}{U^{1/\alpha}} }$

is Pareto-distributed.[38] If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).

## See also另请参阅

Category:Actuarial science

Category:Continuous distributions

Category:Power laws

Category:Probability distributions with non-finite variance

Category:Exponential family distributions

Category:Vilfredo Pareto

This page was moved from wikipedia:en:Pareto distribution. Its edit history can be viewed at Pareto分布/edithistory

1. Amoroso, Luigi (1938). "VILFREDO PARETO". Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest. 6.
2. Pareto, Vilfredo (1898). "Cours d'economie politique". Journal of Political Economy. 6.
3. VAN MONTFORT, M.A.J. (1986). "The Generalized Pareto distribution applied to rainfall depths". Hydrological Sciences Journal. 31 (2): 151–162. doi:10.1080/02626668609491037.
4. Oancea, Bogdan (2017). "Income inequality in Romania: The exponential-Pareto distribution". Physica A: Statistical Mechanics and Its Applications. 469: 486–498. Bibcode:2017PhyA..469..486O. doi:10.1016/j.physa.2016.11.094.
5. Barry C. Arnold (1983). Pareto Distributions. International Co-operative Publishing House. ISBN 978-0-89974-012-6.
6. 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
7. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics. 引用错误：无效<ref>标签；name属性“Johnson1994”使用不同内容定义了多次
8. Johnson, Kotz, and Balakrishnan (1994), (20.4). 引用错误：无效<ref>标签；name属性“jkb94”使用不同内容定义了多次
9. Christian Kleiber & Samuel Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 978-0-471-15064-0.  引用错误：无效<ref>标签；name属性“kk03”使用不同内容定义了多次
10. Feller, W. (1971). An Introduction to Probability Theory and its Applications. II (2nd ed.). New York: Wiley. p. 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 ~ Axα as x → ∞."
11. {cite journal | last1=Lomax | first1=K.S.| year=1954 | title=Business failures。另一个失败数据分析的例子| url=|journal=journal of the American Statistical Association | volume=49 | issue=268 | pages=847–52 | doi=10.1080/01621459.1954.10501239}
12. 引用错误：无效<ref>标签；未给name属性为Feller的引用提供文字
13. Chotikapanich, Duangkamon (16 September 2008). "Chapter 7: Pareto and Generalized Pareto Distributions". Modeling Income Distributions and Lorenz Curves. pp. 121–22. ISBN 9780387727967.
14. Chotikapanich, Duangkamon (16 September 2008). "Chapter 7: Pareto and Generalized Pareto Distributions". Modeling Income Distributions and Lorenz Curves. pp. 121–22. ISBN 9780387727967.
15. http://www.cs.bgu.ac.il/~mps042/invtransnote.htm
16. http://www.cs.bgu.ac.il/~mps042/invtransnote.htm
17. Huang, Xiao-dong (2004). "A Multiscale Model for MPEG-4 Varied Bit Rate Video Traffic". IEEE Transactions on Broadcasting. 50 (3): 323–334. doi:10.1109/TBC.2004.834013.
18. Reed, William J.; et al. (2004). "The Double Pareto-Lognormal Distribution – A New Parametric Model for Size Distributions". Communications in Statistics – Theory and Methods. 33 (8): 1733–53. CiteSeerX 10.1.1.70.4555. doi:10.1081/sta-120037438. Unknown parameter |s2cid= ignored (help)
19. Reed, William J. (2002). "On the rank‐size distribution for human settlements". Journal of Regional Science. 42 (1): 1–17. doi:10.1111/1467-9787.00247. Unknown parameter |s2cid= ignored (help)
20. 引用错误：无效<ref>标签；未给name属性为“Reed”的引用提供文字
21. Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24). "Understanding latent sector error and how to protect against them" (PDF). 8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved 2010-09-10. We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ2 statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.
22. Schroeder, Bianca; Damouras, Sotirios; Gill, Phillipa (2010-02-24). "Understanding latent sector error and how to protect against them" (PDF). 8th Usenix Conference on File and Storage Technologies (FAST 2010). Retrieved 2010-09-10. We experimented with 5 different distributions (Geometric,Weibull, Rayleigh, Pareto, and Lognormal), that are commonly used in the context of system reliability, and evaluated their fit through the total squared differences between the actual and hypothesized frequencies (χ2 statistic). We found consistently across all models that the geometric distribution is a poor fit, while the Pareto distribution provides the best fit.
23. Yuji Ijiri; Simon, Herbert A. (May 1975). "Some Distributions Associated with Bose–Einstein Statistics". Proc. Natl. Acad. Sci. USA. 72 (5): 1654–57. Bibcode:1975PNAS...72.1654I. doi:10.1073/pnas.72.5.1654. PMC 432601. PMID 16578724.
24. Harchol-Balter, Mor; Downey, Allen (August 1997). "Exploiting Process Lifetime Distributions for Dynamic Load Balancing" (PDF). ACM Transactions on Computer Systems. 15 (3): 253–258. doi:10.1145/263326.263344. Unknown parameter |s2cid= ignored (help)
25. Harchol-Balter, Mor; Downey, Allen (August 1997). "Exploiting Process Lifetime Distributions for Dynamic Load Balancing" (PDF). ACM Transactions on Computer Systems. 15 (3): 253–258. doi:10.1145/263326.263344. Unknown parameter |s2cid= ignored (help)
26. [3]
27. Kleiber and Kotz (2003): p. 94.
28. Seal, H. (1980). "Survival probabilities based on Pareto claim distributions". ASTIN Bulletin. 11: 61–71. doi:10.1017/S0515036100006620.
29. Kleiber and Kotz (2003): p. 94.
30. Seal, H. (1980). "Survival probabilities based on Pareto claim distributions". ASTIN Bulletin. 11: 61–71. doi:10.1017/S0515036100006620.
31. CumFreq, software for cumulative frequency analysis and probability distribution fitting [4]
32. CumFreq, software for cumulative frequency analysis and probability distribution fitting [5]
33. Hardy, Michael (2010). "Pareto's Law". Mathematical Intelligencer. 32 (3): 38–43. doi:10.1007/s00283-010-9159-2. Unknown parameter |s2cid= ignored (help)
34. Hardy, Michael (2010). "Pareto's Law". Mathematical Intelligencer. 32 (3): 38–43. doi:10.1007/s00283-010-9159-2. Unknown parameter |s2cid= ignored (help)
35. "62 people own the same as half the world, reveals Oxfam Davos report". Oxfam. Jan 2016.
36. {cite web | title=62人拥有的财富与世界上一半的人相同，展示乐施会达沃斯报告网址=https://www.oxfam.org/en/pressroom/pressreases/2016-01-18/62-people-own-same-half-world-reviews-oxfam-davos-report%7Cpublisher=Oxfam | date=2016年1月}
37. "Global Wealth Report 2013". Credit Suisse. Oct 2013. p. 22. Archived from the original on 2015-02-14. Retrieved 2016-01-24.
38. Tanizaki, Hisashi (2004). Computational Methods in Statistics and Econometrics. CRC Press. p. 133. ISBN 9780824750886.
39. {cite book | last=Tanizaki | first=Hisashi | title=统计学和计量经济学中的计算方法| year=2004 | page=133 | publisher=CRC Press |网址=https://books.google.com/books？id=pOGAUcn13fMC | isbn=9780824750886}