Negative hypergeometric distribution

class ctx.dist_neghypergeo(n, K, N)

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

Let \(A\) and \(B\) be two mutually exclusive events that have already occurred \(v\) and \(w\) times, respectively, in \(v\) + \(w\) trials. Let \(n = k + l\). Then the probability that in the next \(n\) trials events \(A\) and \(B\) will happen \(k\) and \(l\) times, respectively (where \(k\) and \(l\) are nonnegative integers), is

\[\text{pmf}_X(x) = \binom{-v-1}{k} \binom{-w-1}{n-k} \bigg/ \binom{-v-w-2}{n} .\]

The negative-hypergeometric distribution (like the hypergeometric distribution) deals with draws without replacement, so that the probability of success is different in each draw. In contrast, the negative-binomial distribution (like the binomial distribution) deals with draws with replacement, so that the probability of success is the same and the trials are independent. The following table summarizes the four distributions related to drawing items:

Category	With replacements	No replacements
# of successes in constant # of draws	binomial distribution	hypergeometric distribution
# of successes in constant # of failures	negative binomial distribution	negative hypergeometric distribution

See also: Wikipedia [1325], Johnson et al. [411] page 254.

dist_neghypergeo.pmf(x)

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

\[\text{pmf}_X(x) = \binom{-v-1}{k} \binom{-w-1}{n-k} \bigg/ \binom{-v-w-2}{n} .\]

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

dist_neghypergeo.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a negative hypergeometric 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: ", _neghypergeo(mu, sigma).pmf(x))
6.3563523462564525615615615614561356E-20

dist_neghypergeo.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a negative hypergeometric 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: ", _neghypergeo(mu, sigma).pmf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_neghypergeo.qtf(q)

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

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

dist_neghypergeo.isf(q)

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

dist_neghypergeo.g_x(t)

Returns \(G_X(t)\), the characteristic function of a random variable \(X\), following a negative hypergeometric distribution:

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

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: ", _neghypergeo(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_neghypergeo.c_x(t)

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

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

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: ", _neghypergeo(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_neghypergeo.m_x(t)

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

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

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: ", _neghypergeo(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_neghypergeo.k_x(t, k=0)

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

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

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: ", _neghypergeo(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_neghypergeo.moments(k)

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

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

\[\mu'_{[r]} = \frac{n!}{(n-r)!} \frac{a!}{(a-r)!} \frac{1}{(a+b)!}.\]

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: ", _neghypergeo(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_neghypergeo.cumulants(k)

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

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