Hypergeometric: The Hypergeometric Distribution

HypergeometricR Documentation

The Hypergeometric Distribution


Density, distribution function, quantile function and random generation for the hypergeometric distribution.


dhyper(x, m, n, k, log = FALSE)
phyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE)
qhyper(p, m, n, k, lower.tail = TRUE, log.p = FALSE)
rhyper(nn, m, n, k)


x, q

vector of quantiles representing the number of white balls drawn without replacement from an urn which contains both black and white balls.


the number of white balls in the urn.


the number of black balls in the urn.


the number of balls drawn from the urn, hence must be in 0,1,…, m+n.


probability, it must be between 0 and 1.


number of observations. If length(nn) > 1, the length is taken to be the number required.

log, log.p

logical; if TRUE, probabilities p are given as log(p).


logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].


The hypergeometric distribution is used for sampling without replacement. The density of this distribution with parameters m, n and k (named Np, N-Np, and n, respectively in the reference below, where N := m+n is also used in other references) is given by

p(x) = choose(m, x) choose(n, k-x) / choose(m+n, k)

for x = 0, …, k.

Note that p(x) is non-zero only for max(0, k-n) <= x <= min(k, m).

With p := m/(m+n) (hence Np = N \times p in the reference's notation), the first two moments are mean

E[X] = μ = k p

and variance

Var(X) = k p (1 - p) * (m+n-k)/(m+n-1),

which shows the closeness to the Binomial(k,p) (where the hypergeometric has smaller variance unless k = 1).

The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.

In rhyper(), if one of m, n, k exceeds .Machine$integer.max, currently the equivalent of qhyper(runif(nn), m,n,k) is used which is comparably slow while instead a binomial approximation may be considerably more efficient.


dhyper gives the density, phyper gives the distribution function, qhyper gives the quantile function, and rhyper generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is determined by n for rhyper, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.


dhyper computes via binomial probabilities, using code contributed by Catherine Loader (see dbinom).

phyper is based on calculating dhyper and phyper(...)/dhyper(...) (as a summation), based on ideas of Ian Smith and Morten Welinder.

qhyper is based on inversion (of an earlier phyper() algorithm).

rhyper is based on a corrected version of

Kachitvichyanukul, V. and Schmeiser, B. (1985). Computer generation of hypergeometric random variates. Journal of Statistical Computation and Simulation, 22, 127–145.


Johnson, N. L., Kotz, S., and Kemp, A. W. (1992) Univariate Discrete Distributions, Second Edition. New York: Wiley.

See Also

Distributions for other standard distributions.


m <- 10; n <- 7; k <- 8
x <- 0:(k+1)
rbind(phyper(x, m, n, k), dhyper(x, m, n, k))
all(phyper(x, m, n, k) == cumsum(dhyper(x, m, n, k)))  # FALSE
## but error is very small:
signif(phyper(x, m, n, k) - cumsum(dhyper(x, m, n, k)), digits = 3)