Beta-Poisson distribution (Quinkert)

class ctx.dist_betapoisson(\lambda1, a, b)

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

The beta-Poisson distribution is a discrete (lattice) probability distribution . It is a Poisson distribution in which the parameter \(\mu = \lambda_1 p\) where \(\lambda_1\) is a constant and \(p\) is a random variable having a beta distribution with parameters \(a\) and \(b\).

See also Johnson et al. [411] p.368.

dist_betapoisson.pmf(x)

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

\[\text{pmf}_X(x) = \frac{a \cdots (a+x-1) \lambda_1^x}{(a+b) \cdots (a+b+x-1)x!} {}_1F_1(a+x;a+b+x;-\lambda_1), \quad x=0,1, \cdots\]

The following recursion is used for the pmf:

\[(x+2)(x+1) \cdot \text{pmf}_X(x+2) = (x+a+b+\lambda_1)(x+1) \cdot \text{pmf}_X(x+1) - \lambda_1(x+a) \cdot \text{pmf}_X(x).\]

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

dist_betapoisson.cdf(x)

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

\[\text{cdf}_X(x) = \sum_{j=0}^{k} \text{pmf}_X(j).\]

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

dist_betapoisson.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \sum_{j=k+1}^{\infty} \text{pmf}_X(j).\]

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

dist_betapoisson.qtf(q)

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

There is no known closed form for the quantile function \(\text{qtf}_X(x)\): It is computed with the Brent algorithm where the starting values are from a Cornish-Fisher approximation.

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

dist_betapoisson.isf(q)

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

There is no known closed form for the inverse survival function \(\text{isf}_X(x)\): It is computed with the Brent algorithm where the starting values are from a Cornish-Fisher approximation.

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

dist_betapoisson.g_x(t)

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

\[G_X(t) = {}_1F_1(a;a+b; \lambda_1(t-1))\]

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

dist_betapoisson.c_x(t)

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

\[C_X(t) = {}_1F_1(a;a+b; \lambda_1(e^{it}-1))\]

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

dist_betapoisson.m_x(t)

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

\[M_X(t) = {}_1F_1(a;a+b; \lambda_1(e^t-1))\]

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

dist_betapoisson.k_x(t, k=0)

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

\[K_X(t) = \log \left( {}_1F_1(a;a+b; \lambda_1(e^t-1)) \right).\]

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

dist_betapoisson.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an beta-Poisson distribution. The moments are calculated from the factorial moments, which are given by

\[\mu'_{[r]} = \frac{a(a+1) \cdots (a+r-1) \lambda_1^r}{(a+b)(a+b+1) \cdots (a+b+r-1)}\]

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

dist_betapoisson.cumulants(k)

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

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