Hypergeometric {stats}R 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.
m the number of white balls in the urn.
n the number of black balls in the urn.
k the number of balls drawn from the urn.
p probability, it must be between 0 and 1.
nn 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).
lower.tail 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) is given by

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

for x = 0, ..., k.


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.


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.

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.


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)

[Package stats version 2.9.0 Index]