Halphen A distribution

Halphen A distribution#

class ctx.dist_halphen_a(a, b)#

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

The Halphen A distribution is a continuous probability distribution with parameters \(a > 0\) and \(b > 0\), and the support interval \((0, +\infty)\). See also Wikipedia [1243],

See also: Perreault et al. [490], Perreault et al. [491].

dist_halphen_a.pdf(x)#

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

\[\text{pdf}_X(x) = \frac{1}{2 m^{\nu}K_{\nu}(2\alpha)} x^{\nu-1} \exp \left[ -\alpha \left( \frac{x}{m} + \frac{m}{x} \right) \right], x>0.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", fisher_f(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_halphen_a.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Halphen A 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_halphen_a.sf(x)#

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

\[\text{sf}_X(x) = ??\]
>>> 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_halphen_a.qtf(q)#

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

\[\text{qtf}_X(q) = ??\]
>>> 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_halphen_a.isf(q)#

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

\[\text{isf}_X(q) = ??\]
>>> 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_halphen_a.c_x(t)#

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

\[C_X(t) = ??\]
>>> 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_halphen_a.m_x(t)#

Returns NaN, since the moment generating function does not exist.

dist_halphen_a.k_x(t, k=0)#

Returns NaN, since the cumulant generating function does not exist.

dist_halphen_a.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Halphen A distribution. All moments exists.

\[\mu'_X(r) = m^r \frac{K_{\nu+r}(2\alpha)}{K_{\nu}(2\alpha)}.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", fisher_f(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_halphen_a.cumulants(k)#

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

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