Log-series distribution

The following functions return the log-series distribution with mean \(mu\) and the support interval \((0,+\infty)\), and \(0 \le q \le 1\).

See also Wikipedia [1324], MathWorld [901], Johnson et al. [411], Ehrhardt [309] (3.9.17).

class ctx.dist_logseries(p)

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

The zeta distribution is a discrete probability distribution with parameter \(0<p<1\) and support \(k \in \{1,2,3,\cdots \}\)

See also: Wikipedia [1324], MathWorld [901], Johnson et al. [411].

dist_logseries.pmf(x)

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

\[\text{pmf}_X(x) = \frac{-1}{\log(1-p)} \frac{p^k}{k}.\]

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

dist_logseries.cdf(x)

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

\[\text{cdf}_X(x) = 1 + \frac{B(p; k+1, 0)}{\log(1-p)}.\]

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

dist_logseries.sf(x)

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

\[\text{sf}_X(x) = -\frac{B(p; k+1, 0)}{\log(1-p)}\]

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

dist_logseries.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a log-series 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_logseries.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a log-series 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_logseries.g_x(t)

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following a log-series distribution:

\[G_X(t) = \frac{\log(1-p t)}{\log(1-p)}.\]

>>> 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_logseries.c_x(t)

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

\[C_X(t) = \frac{\log(1-p e^{it})}{\log(1-p)}.\]

>>> 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_logseries.m_x(t)

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

\[M_X(t) = \frac{\log(1-p e^t)}{\log(1-p)}, \quad \text{for } t < -\log(p)\]

dist_logseries.k_x(t, k=0)

Returns \(M_X(t)\), the cumulant generating function of a random variable \(X\), following a log-series distribution:

\[K_X(t) = \log \left( \frac{\log(1-p e^t)}{\log(1-p)} \right), \quad \text{for } t < -\log(p)\]

dist_logseries.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a log-series distribution:

\[\mu'_{r} = \frac{\text{Li}_{1-r}(p)}{\log(1-p)}\]

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

dist_logseries.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a log-series 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