Editing Pareto distribution (section)

==Properties==
===Moments and characteristic function===
* The [[expected value]] of a [[random variable]] following a Pareto distribution is
:
:: <math>\operatorname{E}(X)= \begin{cases} \infty & \alpha\le 1, \\
\frac{\alpha x_{\mathrm{m}}}{\alpha-1} & \alpha>1.
\end{cases}</math>
* The [[variance]] of a [[random variable]] following a Pareto distribution is

:: <math>\operatorname{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}</math>

: (If ''α'' ≤ 2, the variance does not exist.)
* The raw [[moment (mathematics)|moments]] are

:: <math>\mu_n'= \begin{cases} \infty & \alpha\le n, \\ \frac{\alpha x_\mathrm{m}^n}{\alpha-n} & \alpha>n. \end{cases}</math>
* The [[Moment-generating function|moment generating function]] is only defined for non-positive values ''t''&nbsp;≤&nbsp;0 as

::<math>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)</math>
::<math>M\left(0,\alpha,x_\mathrm{m}\right)=1.</math>
Thus, since the expectation does not converge on an [[open interval]] containing <math>t=0</math> we say that the moment generating function does not exist.
* The [[Characteristic function (probability theory)|characteristic function]] is given by

:: <math>\varphi(t;\alpha,x_\mathrm{m})=\alpha(-ix_\mathrm{m} t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m} t),</math>

: where Γ(''a'',&nbsp;''x'') is the [[incomplete gamma function]].

The parameters may be solved for using the [[Method of moments (statistics)|method of moments]].<ref>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.</ref>

===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&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>:

:<math>
\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}
</math>

This implies that the conditional expected value (if it is finite, i.e. <math>\alpha>1</math>) is proportional to <math>x_1</math>:

:<math>\text{E}(X | X \geq x_1) \propto x_1.</math>

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.<ref name=":02">{{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 }}</ref>

===A characterization theorem===
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}}

===Geometric mean===
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>

: <math> G = x_\text{m} \exp \left( \frac{1}{\alpha} \right).</math>

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

: <math> H = x_\text{m} \left( 1 + \frac{ 1 }{ \alpha } \right).</math>

===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'' ≥ ''x''<sub>m</sub>,

:<math>\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.</math>

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