Pólya-Eggenberger distribution

class ctx.dist_polya(n, K, N)

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

Suppose that a finite urn initially contains \(w\) white balls and \(b\) black balls and that balls are withdrawn one at a time, with immediate replacement, together with \(c\) balls of a similar color. Then the probability that \(x\) white balls are drawn in a sample of \(n\) withdrawals is

\[\text{pmf}_X(x) = \binom{-w/c}{x} \binom{-b/c}{n-x} \bigg/ \binom{-(w+b)/c}{n} .\]

See also: Johnson et al. [411] page 258.

dist_polya.pmf(x)

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

\[\text{pmf}_X(x) = \binom{-w/c}{x} \binom{-b/c}{n-x} \bigg/ \binom{-(w+b)/c}{n} .\]

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

dist_polya.cdf(x)

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

\[\text{cdf}_X(k) = \sum_{j=\max(0,n+K-N)}^{k} \text{pmf}_X(j) = 1 - \text{pmf}_X(k+1) \times {}_3F_2(1,k+1-K,k+1-n;k+2,N+k+2-K-n;1),\]

where \({}_3F_2(\cdot)\) is a generalized hypergeometric function (see hyp3f2().)

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

dist_polya.sf(x)

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

\[\text{sf}_X(k) = \sum_{j=k+1}^{\min(K,n)} \text{pmf}_X(j) = \text{pmf}_X(k+1) \times {}_3F_2(1,k+1-K,k+1-n;k+2,N+k+2-K-n;1),\]

where \({}_3F_2(\cdot)\) is a generalized hypergeometric function (see hyp3f2().)

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

dist_polya.qtf(q)

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

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

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

\[G_X(t) = \frac{{}_2F_1(-n, w/c; -n+1-b/c; t)}{{}_2F_1(-n, w/c; -n+1-b/c; 1)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

>>> 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_polya.c_x(t)

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

\[C_X(t) = \frac{{}_2F_1(-n, w/c; -n+1-b/c; e^{it})}{{}_2F_1(-n, w/c; -n+1-b/c; 1)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

>>> 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_polya.m_x(t)

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

\[M_X(t) = \frac{{}_2F_1(-n, w/c; -n+1-b/c; e^{t})}{{}_2F_1(-n, w/c; -n+1-b/c; 1)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_polya.k_x(t, k=0)

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

\[K_X(t) = \log \left[ \frac{{}_2F_1(-n, w/c; -n+1-b/c; e^{t})}{{}_2F_1(-n, w/c; -n+1-b/c; 1)} \right].\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_polya.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Pólya-Eggenberger distribution (Wikipedia). The raw moments are calculated from the factorial moments:

\[\mu'_{[r]} = \frac{n! a! (a+b-r)! }{(n-r)! (a-r)! (a+b)!}\]

\[\mu'_{[r]} = \frac{n!}{(n-r)!} \frac{a!}{(a-r)!} \frac{(a+b-r)!}{(a+b)!}.\]

For \(n>0\) and \(a>0\), when \(n \le r\) or \(a \le r\) then \(\mu'_{[r]} = 0\).

When \(a<0\) and \(b<0\) with \(b\) an integer

\[\frac{a!}{(a+b)!} = \frac{(-1)^b (-a-b-1)!}{(-a-1)!}\]

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

dist_polya.cumulants(k)

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