Skellam distribution#

class ctx.dist_skellam(mu1, mu2)#

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

The Skellam distribution is a discrete (lattice) probability distribution of the difference \(X_1-X_2\) of two statistically independent random variables \(X_1\) and \(X_2\), each Poisson-distributed with respective expected values \(\mu_1>0\), \(\mu_2>0\). The support interval is \((-\infty,+\infty)\).

See also: Wikipedia [1329], Jodrá [403].

dist_skellam.pmf(x)#

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

\[\text{pmf}_X(x) = e^{-(\lambda_1+\lambda_2)} \left( \frac{\lambda_1}{\lambda_2} \right)^{k/2} I_k(2\sqrt{\lambda_1+\lambda_2})\]

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

dist_skellam.cdf(x)#

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

\[\text{cdf}_X(x) = \sum_{i=-\infty}^{x} \text{pmf}_X(x)\]

See Jodrá [403] for an expression involving the noncentral \(\chi^2\) distribution

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

dist_skellam.sf(x)#

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

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

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

dist_skellam.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Skellam distribution. There is no closed form for the qtf: It is computed with Newton iterations where the starting values are from Boost.

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

dist_skellam.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a Skellam 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: ", poisson(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_skellam.g_x(t)#

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

\[G_X(t) = \exp( -(\lambda_1+\lambda_2) + \lambda_1 t + \lambda_2 /t ).\]

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

dist_skellam.c_x(t)#

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

\[C_X(t) = \exp( -(\lambda_1+\lambda_2) + \lambda_1 e^{it} + \lambda_2 e^{-it} ).\]

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

dist_skellam.m_x(t)#

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

\[M_X(t) = \exp( -(\lambda_1+\lambda_2) + \lambda_1 e^t + \lambda_2 e^{-t} ).\]

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

dist_skellam.k_x(t, k=0)#

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

\[K_X(t) = -(\lambda_1+\lambda_2) + \lambda_1 e^t + \lambda_2 e^{-t} .\]

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

dist_skellam.moments(k)#

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

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

dist_skellam.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Skellam distribution:

\[\kappa_{2r} = \lambda_1 + \lambda_2,\]

\[\kappa_{2r+1} = \lambda_1 - \lambda_2\]

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

Skellam distribution

Contents

Skellam distribution#