#### 52.2.13 Pareto Random Variable

Function: pdf_pareto (x,a,b)

  Returns the value at x of the density function of a \({\it Pareto}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The pdf is $$ f(x; a, b) = \cases{ \displaystyle{a b^a \over x^{a+1}} & for $x \ge b$ \cr \cr 0 & for $x < b$ } $$

  Categories:Package distrib ·

Function: cdf_pareto (x,a,b)

  Returns the value at x of the distribution function of a \({\it Pareto}(a,b)\) random variable, with a,b>0. To make use of this function, write first `load("distrib")`.

  The cdf is $$ F(x; a, b) = \cases{ 1-\left(\displaystyle{b\over x}\right)^a & for $x \ge b$\cr 0 & for $x < b$ } $$

  Categories:Package distrib ·

Function: quantile_pareto (q,a,b)

  Returns the q-quantile of a \({\it Pareto}(a,b)\) random variable, with a,b>0; in other words, this is the inverse of `cdf_pareto`. Argument q must be an element of [0,1]. To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·

Function: mean_pareto (a,b)

  Returns the mean of a \({\it Pareto}(a,b)\) random variable, with a>1,b>0. To make use of this function, write first `load("distrib")`.

  The mean is $$ E[X] = {ab\over a-1} $$

  Categories:Package distrib ·

Function: var_pareto (a,b)

  Returns the variance of a \({\it Pareto}(a,b)\) random variable, with a>2,b>0. To make use of this function, write first `load("distrib")`.

  The variance is $$ V[X] = {ab^2\over (a-2)(a-1)^2} $$

  Categories:Package distrib ·

Function: std_pareto (a,b)

  Returns the standard deviation of a \({\it Pareto}(a,b)\) random variable, with a>2,b>0. To make use of this function, write first `load("distrib")`.

  The standard deviation is $$ D[X] = {b\over a-1} \sqrt{a\over a-2} $$

  Categories:Package distrib ·

Function: skewness_pareto (a,b)

  Returns the skewness coefficient of a \({\it Pareto}(a,b)\) random variable, with a>3,b>0. To make use of this function, write first `load("distrib")`.

  The skewness coefficient is $$ SK[X] = {2(a+1)\over a-3} \sqrt{a-2\over a} $$

  Categories:Package distrib ·

Function: kurtosis_pareto (a,b)

  Returns the kurtosis coefficient of a \({\it Pareto}(a,b)\) random variable, with a>4,b>0. To make use of this function, write first `load("distrib")`.

  The kurtosis coefficient is $$ KU[X] = {6\left(a^3+a^2-6*a-2\right) \over a(a-3)(a-4)} - 3 $$

  Categories:Package distrib ·

Function: random_pareto (a,b)
`random_pareto` (a,b,n)

  Returns a \({\it Pareto}(a,b)\) random variate, with a>0,b>0. Calling `random_pareto` with a third argument n, a random sample of size n will be simulated.

  The implemented algorithm is based on the general inverse method.

  To make use of this function, write first `load("distrib")`.

  Categories:Package distrib ·Random numbers ·