Zeta distribution

The following functions return class of the Zeta distribution with parameter \(r\), and \(0 \le q \le 1\).

See also Wikipedia [1334], Rinne [505], Johnson et al. [411] page 527, Ehrhardt [309] (3.9.34).

class ctx.dist_zeta(s)

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

The zeta distribution is a discrete probability distribution.

See also: Wikipedia [1334], Rinne [505], Johnson et al. [411] page 527,

dist_zeta.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following a zeta distribution:

\[\text{pmf}_X(x) = \frac{1/k^s}{\zeta(s)}\]

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

dist_zeta.cdf(x)

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

\[\text{cdf}_X(x) = \frac{H_{k,s}}{\zeta(s)}, \quad \text{where } H_{k,s} = \sum_{j=1}^k \frac{1}{j^s} \text{ is the generalized harmonic number}.\]

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

dist_zeta.sf(x)

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

\[\text{sf}_X(x) = 1 - \frac{H_{k,s}}{\zeta(s)}, \quad \text{where } H_{k,s} = \sum_{j=1}^k \frac{1}{j^s} \text{ is the generalized harmonic number}.\]

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

dist_zeta.qtf(q)

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

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

dist_zeta.isf(q)

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

\[\text{isf}_X(q) = \text{qtf}_X(1-q).\]

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

dist_zeta.c_x(t)

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

\[C_X(t) = \frac{\text{Li}_s(e^{it})}{\zeta(s)}\]

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

dist_zeta.m_x(t)

Returns None, since the moment generating function does not exist.

dist_zeta.k_x(t, k=0)

Returns None, since the cumulant generating function does not exist.

dist_zeta.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a zeta distribution (Wikipedia). The moments are finite only for \(r<s-1\) and are then given by

\[\mu'_{r} = \frac{\zeta(s-r)}{\zeta(s)}, \quad \text{for } r < s-1.\]

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

dist_zeta.cumulants(k)

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

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