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 ·