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

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