Negative hypergeometric distribution

In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without replacement in which each sample can be classified into two mutually exclusive categories like Pass/Fail or Employed/Unemployed. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw. Unlike the standard hypergeometric distribution, which describes the number of successes in a fixed sample size, in the negative hypergeometric distribution, samples are drawn until ${\displaystyle r}$ failures have been found, and the distribution describes the probability of finding ${\displaystyle k}$ successes in such a sample. In other words, the negative hypergeometric distribution describes the likelihood of ${\displaystyle k}$ successes in a sample with exactly ${\displaystyle r}$ failures.

Negative hypergeometric

Probability mass function
Cumulative distribution function
Parameters	${\displaystyle N\in \left\{0,1,2,\dots \right\}}$ - total number of elements ${\displaystyle K\in \left\{0,1,2,\dots ,N\right\}}$ - total number of 'success' elements ${\displaystyle r\in \left\{0,1,2,\dots ,N-K\right\}}$ - number of failures when experiment is stopped
Support	${\displaystyle k\in \left\{0,\,\dots ,\,K\right\}}$ - number of successes when experiment is stopped.
PMF	${\displaystyle {\frac {{{k+r-1} \choose {k}}{{N-r-k} \choose {K-k}}}{N \choose K}}}$
Mean	${\displaystyle r{\frac {K}{N-K+1}}}$
Variance	${\displaystyle r{\frac {(N+1)K}{(N-K+1)(N-K+2)}}[1-{\frac {r}{N-K+1}}]}$