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In probability theory and statistics, the negative hypergeometric distribution describes the probability of the number of elements taken without replacement from a finite population whose elements can be classified into two mutually exclusive categories like Pass/Fail, Male/Female or Employed/Unemployed that stops when a fixed number of elements of certain class have been taken. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.

Negative hypergeometric Parameters

Parameters	\(N \in \left\{0,1,2,\dots\right\} \) - total number of elements \(K \in \left\{0,1,2,\dots,N\right\} \) - total number of 'success' elements \( R \in \left\{0,1,2,\dots,N-K\right\}\) - number of failures when experiment is stopped
Support	\(k \in \left\{0,\, \dots,\, K\right\} \) - number of successes when experiment is stopped.
Mean	\( R\frac{K}{N-K+1}\)
Variance	\(R\frac{(N+1)K}{(N-K+1)(N-K+2)}[1-\frac{R}{N-K+1}] \)

Definition

There are N elements, of which K are defined as "successes" and the rest are "failures".

Elements are drawn one after the other, without replacements, until R failures are encountered. Then, the drawing stops and the number k of successes is counted. The negative hypergeometric distribution, \( NHG_{N,K,R}(k) \) is the discrete distribution of this k.

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Related distributions

If the drawing stops after a constant number n of draws (regardless of the number of failures), then the number of successes has the hypergeometric distribution, HG_{N,K,n}(k). The two functions are related in the following way:[1]

NHG_{N,K,R}(k) = 1-HG_{N,N-K,k}(R-1)

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, 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:

	With replacements	No replacements
Constant number of draws	binomial distribution	hypergeometric distribution
Constant number of failures	negative binomial distribution	negative hypergeometric distribution

References

Negative hypergeometric distribution in Encyclopedia of Math.