Negative hypergeometric distribution

The negative hypergeometric distribution is a probability distribution on a finite support. It belongs to the univariate discrete probability distributions and can be derived from the urn model .

definition

A random variable on the carrier is called negatively hypergeometrically distributed if it has the probability function ${\ displaystyle X}$ ${\ displaystyle T = \ {k, \ dots, k + NM \}}$

{\ displaystyle f (n) = {\ frac {{\ binom {n-1} {k-1}} \ cdot {\ binom {Nn} {Mk}}} {\ binom {N} {M}}} = {\ frac {{\ binom {-k} {nk}} \ cdot {\ binom {kM-1} {NM-n + k}}} {\ binom {-M-1} {NM}}}}

Has. It is . Then you write . ${\ displaystyle k \ leq M \ leq N}$ ${\ displaystyle X \ sim \ operatorname {NH} (N, M, k)}$

Derivation from the urn model

The negative hypergeometric distribution arises elementarily from the urn model. If one looks at an urn with balls, of which are marked, and pulls out of this urn without putting them back until one has drawn marked balls, the probability of needing draws for this is distributed negatively hypergeometrically. ${\ displaystyle N}$ ${\ displaystyle M}$ ${\ displaystyle k}$ ${\ displaystyle n}$

If you think of drawing all balls one after the other from the urn, then there are a total of possibilities to distribute the marked balls over the draws. The event that exactly in the -th move the -th marked ball is drawn occurs exactly when marked balls are drawn in the moves before and the remaining marked balls are drawn in the moves after . There are ways to do this . ${\ displaystyle N}$ ${\ displaystyle {\ tbinom {N} {M}}}$ ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle n}$ ${\ displaystyle k}$ ${\ displaystyle n-1}$ ${\ displaystyle k-1}$ ${\ displaystyle Nn}$ ${\ displaystyle Mk}$ ${\ displaystyle {\ tbinom {n-1} {k-1}} \ cdot {\ tbinom {Nn} {Mk}}}$