# Hypergeometric

0th

Percentile

##### The Hypergeometric Distribution

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

Keywords
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.
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 $P[X \le x]$, 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 $$p(x) = \left. {m \choose x}{n \choose k-x} \right/ {m+n \choose k}%$$ 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. If one of $m, n, k$, exceeds .Machine\$integer.max, currently the equivalent of qhyper(runif(nn), m,n,k) is used, when 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.

##### 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.

##### Aliases
• Hypergeometric
• dhyper
• phyper
• qhyper
• rhyper
##### Examples
library(stats) 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) 
Documentation reproduced from package stats, version 3.3.3, License: Part of R 3.3.3

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