Boost: Gumbel (Generalized Extreme Value distribution Type-I) distribution

The following functions return the pdf, cdf, qtf or boost class of the Gumbel distribution with parameters \(a \in \mathbb{R}\) (location), \(b > 0\) (scale), and the support interval \((-\infty, +\infty)\).

See also Wikipedia [1245], MathWorld [895], BoostMath [61], Ehrhardt [309] (3.9.8).

Ctx.gumbel_pdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

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

\[\text{pdf}(x) = \frac{e^{-(x-a)/b}}{b} e^{-e^{-(x-a)/b}}.\]

The following example shows both forms of the syntax:

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

Ctx.gumbel_cdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

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

\[\text{cdf}(x) = e^{-e^{-(x-a)/b}}.\]

The following example shows both forms of the syntax:

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

Ctx.gumbel_qtf(q, a=0, b=1)

where Ctx is Math53 or CtxBoost.

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

\[\text{qtf}(q) = a - b \log(-\log(q)).\]

The following example shows both forms of the syntax:

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

class ctx.dist_gumbel(a, b)

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

The Gumbel (or Extreme Value Type I )distribution is a continuous probability distribution with parameters \(a \in \mathbb{R}\) (location), \(b > 0\) (scale), and the support interval \((-\infty, +\infty)\).

See also Wikipedia [1245], MathWorld [895], BoostMath [61], Witkovský [1619].

Note: In MathWorld/Mathematica, the Gumbel (Maximum) distribution is called the ExtremeValueDistribution, and the Gumbel (Minimum) distribution is called the GumbelDistribution.

dist_gumbel.pdf(x)

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

\[\text{pdf}_X(x) = \frac{e^{-(x-a)/b}}{b} e^{e^{-(x-a)/b}}.\]

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

dist_gumbel.cdf(x)

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

\[\text{cdf}_X(x) = e^{e^{-(x-a)/b}}.\]

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

dist_gumbel.sf(x)

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

\[\text{sf}_X(x) = 1 - e^{e^{-(x-a)/b}} = -\text{expm1}(-e^{-(x-a)/b}).\]

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

dist_gumbel.qtf(q)

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

\[\text{qtf}_X(q) = a - b \log(-\log(q)).\]

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

dist_gumbel.isf(q)

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

\[\text{isf}_X(q) = a - b \log(-\text{log1p}(-q)).\]

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

dist_gumbel.c_x(t)

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

\[C_X(t) = \Gamma(1 - i b t) e^{i a t}.\]

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

dist_gumbel.m_x(t)

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

\[M_X(t) = \Gamma(1 - b t) e^{a t}.\]

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

dist_gumbel.k_x(t, k=0)

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

\[K_X(t) = a t + \log (\Gamma(1 - b t)).\]

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

dist_gumbel.moments(k)

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

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

dist_gumbel.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an Gumbel distribution:

\[\kappa'_{X}(n) = (n-1)! \zeta(n).\]

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