| Hypergeometric {stats} | R Documentation |
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)
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, hence must be in
|
p |
probability, it must be between 0 and 1. |
nn |
number of observations. If |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
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, where N := m+n is also used
in other references) is given by
p(x) = \left. {m \choose x}{n \choose k-x} \right/ {m+n \choose k}%
for x = 0, \ldots, k.
Note that p(x) is non-zero only for
\max(0, k-n) \le x \le \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] = \mu = k p
and variance
\mbox{Var}(X) = k p (1 - p) \frac{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) \ge 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.
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 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.
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 (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.
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 <- 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 errors are very small:
signif(phyper(x, m, n, k) - cumsum(dhyper(x, m, n, k)), digits = 3)
stopifnot(abs(phyper(x, m, n, k) - cumsum(dhyper(x, m, n, k))) < 5e-16)