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.. _rst_dist_polya: 

Pólya-Eggenberger distribution
===============================================================================


.. py: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    Suppose that a finite urn initially contains `w` white balls and `b` black balls and that balls are withdrawn one at a time, with immediate replacement, together with `c` balls of a similar color.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

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

    See also: :cite:t:`Johnson2005` page 258.




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.. method:: dist_polya.pmf(x)

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

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



    .. code-block:: python

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



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.. method:: dist_polya.cdf(x)

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

    .. math:: \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  :ref:`hyp3f2() <rst_mpm_hyp3f2>`.)



    .. code-block:: python

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




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.. method:: dist_polya.sf(x)

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

    .. math:: \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  :ref:`hyp3f2() <rst_mpm_hyp3f2>`.)


    .. code-block:: python

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



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.. method:: dist_polya.qtf(q)

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



    .. code-block:: python

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




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.. method:: dist_polya.isf(q)

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

    .. math:: \text{isf}_X(q) = \text{qtf}_X(1-q).


    .. code-block:: python

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




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.. method:: dist_polya.g_x(t)

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

    .. math::  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  :ref:`hyp2f1() <rst_mpm_hyp2f1>`.)



    .. code-block:: python

        >>> 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




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.. method:: dist_polya.c_x(t)

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

    .. math::  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  :ref:`hyp2f1() <rst_mpm_hyp2f1>`.)



    .. code-block:: python

        >>> 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




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.. method:: dist_polya.m_x(t)

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

    .. math::  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  :ref:`hyp2f1() <rst_mpm_hyp2f1>`.)


    .. code-block:: python

        >>> 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




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.. method:: 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:

    .. math:: 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  :ref:`hyp2f1() <rst_mpm_hyp2f1>`.)



    .. code-block:: python

        >>> 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









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.. method:: 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:


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


    .. math::  \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

    .. math::  \frac{a!}{(a+b)!} = \frac{(-1)^b (-a-b-1)!}{(-a-1)!}



    .. code-block:: python

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



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.. method:: 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.


    .. code-block:: python

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