Pareto random variable

$X$ is a Pareto random variable with parameters $a,b$ if

$f_X(x)=\frac{a{b}^a}{x^{a+1}}$, $x\in [b,\mathrm{\infty })$

Parameters:

• $a\in (0,\mathrm{\infty })$

• $b\in (0,\mathrm{\infty })$

Syntax:

$X\sim Pareto(a,b)$

Notes:

1. $X$ represents a random variable with shape parameter $a$ and scale parameter $b$.

2. The expected value of $X$ is noted as $E[X]=\frac{ab}{a-1}$ with $a\in \{2,3,\mathrm{\dots }\}$

3. The variance of $X$ is noted as $Var[X]=\frac{a{b}^2}{{(a-1)}^2(a-2)}$, $a\in \{3,4,\mathrm{\dots }\}$

4. The cumulative distribution function of $X$ is noted as $F(x)=1-{(\frac{b}{x})}^a$

5. The moments of $X$ around 0 are noted as $E[X^n]=\frac{a{b}^n}{a-n}$, $n\in \{1,2,\mathrm{\dots },a-1\}$

Title	Pareto random variable
Canonical name	ParetoRandomVariable
Date of creation	2013-03-22 12:34:05
Last modified on	2013-03-22 12:34:05
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	9
Author	alozano (2414)
Entry type	Definition
Classification	msc 62E15
Synonym	Pareto distribution