Reference: The Hypergeometric Distribution

Description

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

Usage

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)

dhper    计算某一个点的p值
phyper   计算一个分布的p值，默认下计算P(X<=x)
qhyper   计算某一个p值对应的分位数
rhyper   按超几何分布给出nn的可能的模拟结果值

Arguments

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].

Details

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.

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

Value

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 parameters for the other functions.

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

Source

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.

References

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.

Examples

m x 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)