Editing Pareto distribution (section)

===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>