Meixner distribution

class ctx.dist_meixner(n1, n2, lambda, **kwargs)

These functions return PDF, CDF, and ICDF of the Meixner distribution with location \(m \in \mathbb{R}\), scale \(a > 0\), shape parameters \(b\) and \(d\) with \(-\pi<b<\pi\), \(d>0\) and the support interval \((-\infty,+\infty)\) .

See also: MathWorld [260], Grigoletto and Provasi [369].

dist_meixner.pdf(x)

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

\[\text{pdf}_X(x) = \frac{(2\cos(b/2))^{2d}}{2\pi\Gamma(2d)} e^{bz} |\Gamma(d+iz )|^2, \quad \text{where } z=\frac{x-m}{a}.\]

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

dist_meixner.cdf(x)

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

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

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

dist_meixner.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x).\]

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

dist_meixner.qtf(q)

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

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

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

dist_meixner.isf(q)

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

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

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

dist_meixner.c_x(t)

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

\[C_X(t) = e^{m \cdot it} \left( \frac{2\cos(b/2)}{\cosh((at-ib)/2)} \right)^{2d}\]

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

dist_meixner.m_x(t)

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

\[M_X(t) = ??\]

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

dist_meixner.k_x(t, k=0)

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

\[K_X(t) = ??\]

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

dist_meixner.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Meixner distribution (Wikipedia). The moments of the Meixner distribution are usually calculated as the moments of the complex exponential z = eix rather than the angle x itself. These moments are referred to as circular moments. The variance calculated from these moments is referred to as the circular variance.

\[??\]

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

dist_meixner.cumulants(k)

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

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