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当在线性轴上绘制时，分布曲线为熟悉的J形曲线，该曲线[[渐近]]地接近每个正交轴。曲线的所有段都是自相似的（取决于适当的比例因子）。在[[双对数图]]中绘制时，分布用直线表示。

当在线性轴上绘制时，分布曲线为熟悉的J形曲线，该曲线[[渐近]]地接近每个正交轴。曲线的所有段都是自相似的（取决于适当的比例因子）。在[[双对数图]]中绘制时，分布用直线表示。

==Properties属性==

==Properties性质==

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

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

* The [[Characteristic function (probability theory)|characteristic function]] is given by

* The [[Characteristic function (probability theory)|characteristic function]] is given by

*[[特征函数（概率论）|特征函数]]由

*[[特征函数（概率论）|特征函数]]由以下给出

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

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

其中 γ (a，x)是不完全Γ函数。

其中 Γ(a, x)是不完全Γ函数。

The parameters may be solved using the method of moments.<!-- :

The parameters may be solved using the method of moments.<!-- :

参数可以用矩量法求解。 < ! -- :

参数可以用<font color="#ff8000"> 矩量法Method of moments</font>求解。 < ! -- :

alpha = 1 + sqr(1 + mean ^ 2 / var)

alpha = 1 + sqr(1 + mean ^ 2 / var)

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

一个帕雷托分布的随机变量的条件概率分布，假设它大于或等于一个特定的数字 x 1超过 x _ text { m } ，是一个具有相同帕雷托指数 α 但最小 x _ 1而不是 x _ text { m }的帕累托分布。

一个Pareto分布随机变量的条件概率分布，如果它大于或等于某个特定的数&nbsp;x_1 exceeding x_\text{m}，则它是具有相同Pareto指数的Pareto分布，但是具有最小的&nbsp;x_1而不是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&nbsp;<math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index&nbsp;<math>\alpha</math> but with minimum&nbsp;<math>x_1</math> instead of <math>x_\text{m}</math>.

The [[conditional probability distribution]] of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number&nbsp;<math>x_1</math> exceeding <math>x_\text{m}</math>, is a Pareto distribution with the same Pareto index&nbsp;<math>\alpha</math> but with minimum&nbsp;<math>x_1</math> instead of <math>x_\text{m}</math>.

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 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.

假设 x _ 1，x _ 2，x _ 3，dotsc 是独立同分布的随机变量，对于某些 x _ text { m } > 0，其概率分布在区间[ x _ text { m } ，infty ]上是支持的。假设对于所有 n，两个随机变量 min { x1，dotsc，xn }和(x1 + dotsb + xn)/min { x1，dotsc，xn }是独立的。那么公共分配就是帕累托分布。

假设X_1, X_2, X_3, \dotsc是独立同分布的随机变量，其概率分布在区间 [x_\text{m},\infty)上，对于某些x_\text{m}>0成立。假设对于所有n，两个随机变量 \min\{X_1,\dotsc,X_n\}和(X_1+\dotsb+X_n)/\min\{X_1,\dotsc,X_n\} 相互独立，那么其公共分布就是<font color="#ff8000"> 帕累托分布</font>。

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

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

The geometric mean (G) is

The geometric mean (G) is

几何平均(g)是

<font color="#ff8000"> 几何平均数(G)</font>是

The [[geometric mean]] (''G'') is<ref name=Johnson1994>Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.</ref>

The [[geometric mean]] (''G'') is<ref name=Johnson1994>Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.</ref>

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 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.

调和平均数(h)是帕累托分布，称为帕累托 i 型、 II 型、 III 型、 IV 型和 Feller-帕累托分布。帕累托类型 IV 包含帕累托类型 i-III 作为特殊情况。Feller-帕累托分布推广了 Pareto 第四型。

<font color="#ff8000"> 调何平均数(H)</font>是帕累托分布，称为帕累托 i 型、 II 型、 III 型、 IV 型和 Feller-帕累托分布。帕累托类型 IV 包含帕累托类型 i-III 作为特殊情况。Feller-帕累托分布推广了 Pareto 第四型。

The [[harmonic mean]] (''H'') is<ref name="Johnson1994"/>

The [[harmonic mean]] (''H'') is<ref name="Johnson1994"/>

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'' ≥ ''x''_m,

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'' ≥ ''x''_m,

当在线性标度上绘制时，特征曲线“[[长尾]]”分布在[[对数曲线图]]上绘制时，掩盖了函数潜在的简单性，然后采用负梯度的直线形式：根据概率密度函数的公式，对于“x”≥“x”_m，

当在线性标度上绘制时，“[[长尾]]”分布特征曲线在[[对数曲线图]]上绘制时，掩盖了函数潜在的简单性，然后采用负梯度的直线形式：根据概率密度函数的公式，对于''x'' ≥ ''x''_m，

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

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

Since ''α'' is positive, the gradient −(''α''&nbsp;+&nbsp;1) is negative.

Since ''α'' is positive, the gradient −(''α''&nbsp;+&nbsp;1) is negative.

由于“α”为正，因此梯度−（“”α“+&nbsp；1）为负。

由于“α”为正，因此梯度−(''α''&nbsp;+&nbsp;1)为负。

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