Density, distribution function, quantile function and random generation for the hypergeometric distribution.
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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 0,1,…, m+n. |
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 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.
Distributions for other standard distributions.
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