Boost: Pareto distribution

The following functions return pdf, cdf, qtf or boost class of the Pareto distribution with minimum (real) value \(k > 0\), shape \(a > 0\), and the support interval \((k, +\infty)\).

See also Wikipedia [1259], MathWorld [907], BoostMath [74], Ehrhardt [309] (3.9.25).

Ctx.pareto_pdf(x, k, a)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the Pareto distribution:

\[\text{pdf}(x) = \frac{a}{x} \left(\frac{k}{x}\right)^a.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ParetoPdf(x, a, b): ", ParetoPdf(x, a, b))
>>> print ("dist_pareto(a, b).pdf(x): ", dist_pareto(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.pareto_cdf(x, k, a)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the Pareto distribution:

\[\text{cdf}(x) = 1 - \left(\frac{k}{x}\right)^a = - \text{powm1}(k/x,a).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ParetoCdf(x, a, b): ", ParetoCdf(x, a, b))
>>> print ("dist_pareto(a, b).cdf(x): ", dist_pareto(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.pareto_qtf(q, k, a)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Pareto distribution:

\[\text{qtf}(q) = \frac{k}{(1-q)^{1/a}}.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("ParetoQtf(q, a, b): ", ParetoQtf(q, a, b))
>>> print ("dist_pareto(a, b).qtf(q): ", dist_pareto(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_pareto(k, a)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The Pareto distribution is a continuous probability distribution with minimum (real) value \(k > 0\), shape \(a > 0\), and the support interval \((k, +\infty)\). See also Wikipedia [1259], MathWorld [907], BoostMath [74], Witkovský [1627].

dist_pareto.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Pareto distribution:

\[\text{pdf}_X(x) = \frac{a}{x} \left(\frac{k}{x}\right)^a.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", pareto(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_pareto.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Pareto distribution:

\[\text{cdf}_X(x) = 1 - \left(\frac{k}{x}\right)^a = - \text{powm1}(k/x,a).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", pareto(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_pareto.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following an Pareto distribution:

\[\text{sf}_X(x) = \left(\frac{k}{x}\right)^a = \text{pow}(k/x,a).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", pareto(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_pareto.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following an Pareto distribution:

\[\text{qtf}_X(q) = \frac{k}{(1-q)^{1/a}}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", pareto(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_pareto.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following an Pareto distribution:

\[\text{isf}_X(q) = \frac{k}{q^{1/a}}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", pareto(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_pareto.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an Pareto distribution:

\[C_X(t) = a(-ikt)^a \Gamma(-a, -ikt).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", pareto(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_pareto.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an Pareto distribution:

\[M_X(t) = a(-kt)^a \Gamma(-a, -kt), \quad \text{for } t<0.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("m_x: ", pareto(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_pareto.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Pareto distribution:

\[K_X(t) = K_X(t) = \log(M_X(t)), \quad \text{for } t<0.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", pareto(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_pareto.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Pareto distribution. The rth moments only exists for \(n_2 > 2r\).

\[\mu'_{X}(r) = \frac{a k}{a - n} \quad \text{for } a>n.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", pareto(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_pareto.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an Pareto distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", pareto(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00