Boost: Poisson distribution

Boost: Poisson distribution#

The following functions return the pmf, cdf, qtf or boost class of the Poisson distribution with mean \(mu\) and the support interval \((0,+\infty)\), and \(0 \le q \le 1\).

See also Wikipedia [1281], MathWorld [908], BoostMath [90], Ehrhardt [309] (3.9.26).

Ctx.poisson_pmf(k, mu)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pmf}(x)\), the value of the probability mass function (Pmf) of the Poisson distribution:

\[\text{pmf}(x) = \frac{\mu^k}{k!} e^{-\mu}.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("PoissonPdf(x, a, b): ", PoissonPdf(x, a, b))
>>> print ("dist_poisson(a, b).pdf(x): ", dist_poisson(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.poisson_cdf(k, mu)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the Poisson distribution:

\[\text{cdf}(x) = e^{-\mu} \sum_{i=0}^k \frac{\mu^i}{i!} = Q(1+k,\mu).\]

Here \(Q(\cdot)\) denotes the upper regularized incomplete gamma function (RealGammaQ).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("PoissonCdf(x, a, b): ", PoissonCdf(x, a, b))
>>> print ("dist_poisson(a, b).cdf(x): ", dist_poisson(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.poisson_qtf(q, mu)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Poisson distribution:

\[\text{qtf}(q) = \mathrm{gammaq\_inva}(\mu, p) - 1.\]

Here \(\mathrm{gammaq\_inva}(\cdot)\) denotes the inverse (on parameter \(a\)) of the real upper normalised incomplete gamma function (RealGammaQInva).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("PoissonQtf(q, a, b): ", PoissonQtf(q, a, b))
>>> print ("dist_poisson(a, b).qtf(q): ", dist_poisson(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_poisson(lambda1)#

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

The Poisson distribution is a discrete (lattice) probability distribution with mean \(mu\) and the support interval \((0,+\infty)\). See also Wikipedia [1281], MathWorld [908], BoostMath [90], and Witkovský [1644], R (Statistical System) [560].

dist_poisson.pmf(x)#

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

\[\text{pmf}_X(x) = \frac{\lambda_1^k}{k!} e^{-\lambda_1}.\]

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

dist_poisson.cdf(x)#

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

\[\text{cdf}_X(x) = e^{-\lambda_1} \sum_{i=0}^k \frac{\lambda_1^i}{i!} = Q(1+k,\lambda_1).\]

Here \(Q(\cdot)\) denotes the upper regularized incomplete gamma functions.

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

dist_poisson.sf(x)#

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

\[\text{sf}_X(x) = 1 - e^{-\lambda_1} \sum_{i=0}^k \frac{\lambda_1^i}{i!} = P(1+k,\lambda_1).\]

Here \(P(\cdot)\) denotes the lower regularized incomplete gamma functions.

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

dist_poisson.qtf(q)#

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

\[\text{qtf}_X(q) = \mathrm{gamma_q\_inva}(\lambda_1, p) - 1.\]

Here \(\mathrm{gamma_q\_inva}(\cdot)\) denotes the inverse (on parameter \(a\)) of the real normalised incomplete gamma function.

>>> 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_poisson.isf(q)#

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

\[\text{isf}_X(q) = \mathrm{gamma_p\_inva}(\lambda_1, q) - 1.\]

Here \(\mathrm{gamma_p\_inva}(\cdot)\) denotes the inverse (on parameter \(a\)) of the real normalised complementary incomplete gamma function.

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

dist_poisson.g_x(t)#

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

\[G_X(t) = \exp( \lambda(t -1)).\]

>>> 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_poisson.c_x(t)#

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

\[C_X(t) = \exp( \lambda(e^{it} -1)).\]

>>> 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_poisson.m_x(t)#

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

\[M_X(t) = \exp( \lambda(e^{t} -1)).\]

\[L_X(t) = \exp( \lambda(e^{-t} -1)).\]

>>> 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_poisson.k_x(t, k=0)#

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

\[K_X(t) = \lambda(e^t -1).\]

>>> 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_poisson.moments(k)#

Returns the first \(j\) moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Poisson 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_poisson.cumulants(k)#

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

\[\kappa_r = \lambda, \quad r = 1, 2, ...\]

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

Approximations

ctx.poisson_ecf(x, f, rho, omega)#

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

Calculates the Edgeworth approximation to the pdf, cdf and sf.

ctx.poisson_ecf_inv(x, f, rho, omega)#

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

Calculates the Cornish-Fisher approximation to the qtf and isf.

ctx.poisson_spa(x, n, results='c')#

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

Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf.

The solution \(\hat{s}(x)\) of the saddlepoint equation \(K_X^{(1)}(\hat{s}(x))=x\), of a random variable \(X\), following a Poisson distribution is given by:

\[\hat{s}(x)= \log ( k / \lambda).\]

ctx.poisson_spa_inv(x, n, results='qtf')#

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

Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.

Boost: Poisson distribution

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Boost: Poisson distribution#