Negative hypergeometric distribution

Negative hypergeometric
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	$R{\frac {(N+1)K}{(N-K+1)(N-K+2)}}[1-{\frac {R}{N-K+1}}]$

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, Male/Female 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 $R$ failures have been found, and the distribution describes the probability of finding $k$ successes in such a sample. In other words, the negative hypergeometric distribution describes the likelihood of $k$ successes in a sample with exactly $R$ failures.

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 $k$ .

^[1]

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

1 2 Negative hypergeometric distribution in Encyclopedia of Math.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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