Delaporte distribution

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

These functions return PDF, CDF, and ICDF of the Delaporte distribution with location \(a\), scale \(b > 0\), and the support interval \((-\infty,+\infty)\) :

See also Wikipedia [1315], Witkovský [1636].

dist_delaporte.pmf(x)

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

\[\text{pmf}_X(x) = \sum_{i=0}^k\frac{\Gamma(\alpha + i)\beta^i\lambda^{k-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(k-i)!}\]

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

dist_delaporte.cdf(x)

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

\[\text{cdf}_X(x) = \sum_{j=0}^k\sum_{i=0}^j\frac{\Gamma(\alpha + i)\beta^i\lambda^{j-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(j-i)!}\]

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

dist_delaporte.sf(x)

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following an Delaporte 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_delaporte(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_delaporte.qtf(q)

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

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

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

dist_delaporte.isf(q)

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

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

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

dist_delaporte.c_x(t)

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

\[C_X(t) = ??\]

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

dist_delaporte.m_x(t)

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

\[M_X(t) = \frac{e^{\lambda(e^{t}-1)}}{(1-\beta(e^{t}-1))^\alpha}\]

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

dist_delaporte.k_x(t, k=0)

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

\[K_X(t) = ??\]

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

dist_delaporte.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Delaporte distribution (Wikipedia). The raw moments are calculated from the central moments.

\[\mu_{X}(r) = ??\]

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

dist_delaporte.cumulants(k)

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

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