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.