Generalized beta-exponential distribution

class ctx.dist_gen_beta_exp(a, b)

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

The generalized beta-exponential distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), and the support interval \((0, +\infty)\).

See also: Wikipedia [1292], Kleiber and Kotz [415] page 184, equ. 6.6, distribution of Y = log(X), Ristić et al. [506].

This is the distribution of the generalized logistic distribution, for b = a = 1.

dist_gen_beta_exp.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a generalized beta-exponential distribution:

\[\text{pdf}_X(x) = \frac{a e^{ap(x-\log(b))}}{B(p,q) [1+e^{a(x-\log(b))} ]^{p+q}}, \quad -\infty<x<\infty.\]

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

dist_gen_beta_exp.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a generalized beta-exponential distribution:

\[\text{cdf}_X(x) = ??\]

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

dist_gen_beta_exp.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a generalized beta-exponential distribution:

\[\text{sf}_X(x) = ??\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function, and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function.

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

dist_gen_beta_exp.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a generalized beta-exponential distribution:

\[\text{qtf}_X(q) = ??\]

Here \(\mathrm{ibeta\_inv}(\cdot)\) denotes the inverse of the real normalised incomplete beta function.

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

dist_gen_beta_exp.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a generalized beta-exponential distribution:

\[\text{isf}_X(q) = ??\]

Here \(\mathrm{ibetac\_inv}(\cdot)\) denotes the inverse of the real normalised complementary incomplete beta function.

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

dist_gen_beta_exp.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a generalized beta-exponential distribution:

\[C_X(t) = \frac {\Gamma (\beta - it)\Gamma (\alpha + it)}{\Gamma (\alpha )\Gamma (\beta )}\]

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

dist_gen_beta_exp.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a generalized beta-exponential distribution:

\[M_X(t) = b^t \frac {\Gamma (p+t/a)\Gamma (q-t/a)}{\Gamma (p)\Gamma (q)},\quad -p <t<q.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).moments(k))
6.3563523462564525625625625624562356E+00

dist_gen_beta_exp.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a generalized beta-exponential distribution:

\[K_X(t) = \log(\Gamma (\beta -t)) + \log(\Gamma (\alpha +t)) - \log(\Gamma (\alpha )) - \log(\Gamma (\beta )),\quad -\alpha <t<\beta .\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 2; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).moments(k))
6.3563523462564525625625625624562356E+00

dist_gen_beta_exp.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 2 \ldots k\), of a random variable \(X\), following a generalized beta-exponential distribution. The moments are calculated from the cumulants.

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

dist_gen_beta_exp.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 2 \ldots k\), of a random variable \(X\), following a generalized beta-exponential distribution. The cumulants are given by

\[\kappa_X(r) = \psi^{r-1}(\alpha) + (-1)^r \psi^{r-1}(\beta)\]

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