Pareto random variable

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

$f_{X}(x)=\frac{ab^{a}}{x^{a+1}}$, $x\in[b,\infty)$

Parameters:

• $\star$

$a\in(0,\infty)$

• $\star$

$b\in(0,\infty)$

Syntax:

$X\sim Pareto(a,b)$

Notes:

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

Title Pareto random variable ParetoRandomVariable 2013-03-22 12:34:05 2013-03-22 12:34:05 alozano (2414) alozano (2414) 9 alozano (2414) Definition msc 62E15 Pareto distribution