Hypergeometric distributions
hypergeometric_distribution
Definition
\(X \sim Hyp(m_1, m_2, k)\) if
\[ f_X(\ell) = {{m_1 \choose \ell} {m_2 \choose k - \ell} \over {m_1 + m_2 \choose k}} \]The support is
\[ 0 \vee (k - m_2) \le \ell \le m_1 \wedge k \]The corresponding orthogonal polynomial is the Hahn polynomial (see e.g. (2.4.9) of [{nikiforov-suslov-uvarov91}]).
q-analog
A \(q\)-deformation is: \(X \sim qHyp(m_1, m_2, k)\) if
\[ f_X(\ell) = q^{(m_1 - \ell)(k - \ell)}{{m_1 \choose \ell}_q {m_2 \choose k - \ell}_q \over {m_1 + m_2 \choose k}_q} \]See (6) of q_pochhammers for a proof that it is a probability distribution.
properties:
- When \(q \to 0\), \(f_X (\ell) = \ind_{\ell = m_1 \wedge k}\).
- When \(q \to 1\), \(qHyp \to Hyp\) with the same parameters.
- When \(k = 0\), \(f_X (\ell) = \ind_{\ell = 0}\).
- When \(k = \infty\) \[ f_X (\ell) = q^{(m_1 - \ell) (k - \ell)} {(m_1)_q (k)_q \over (\ell)_q (m_1 - \ell)_q (k - \ell)_q} \]
References
- [nikiforov-suslov-uvarov91] Classical orthogonal polynomials of a discrete variable, , 1991.