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% File src/library/stats/man/Hypergeometric.Rd
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% File src/library/stats/man/Hypergeometric.Rd
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% Part of the R package, https://www.R-project.org
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% Part of the R package, https://www.R-project.org
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% Copyright 1995-2014 R Core Team
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% Copyright 1995-2016 R Core Team
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% Distributed under GPL 2 or later
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% Distributed under GPL 2 or later
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\name{Hypergeometric}
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\name{Hypergeometric}
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\alias{Hypergeometric}
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\alias{Hypergeometric}
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\alias{dhyper}
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\alias{dhyper}
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white balls.}
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white balls.}
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\item{m}{the number of white balls in the urn.}
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\item{m}{the number of white balls in the urn.}
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\item{n}{the number of black balls in the urn.}
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\item{n}{the number of black balls in the urn.}
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\item{k}{the number of balls drawn from the urn.}
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\item{k}{the number of balls drawn from the urn.}
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\item{p}{probability, it must be between 0 and 1.}
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\item{p}{probability, it must be between 0 and 1.}
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\item{nn}{number of observations. If \code{length(nn) > 1}, the length
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\item{nn}{number of observations. If \code{length(nn) > 1}, the length
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is taken to be the number required.}
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is taken to be the number required.}
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\item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).}
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\item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).}
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\item{lower.tail}{logical; if TRUE (default), probabilities are
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\item{lower.tail}{logical; if TRUE (default), probabilities are
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\eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.}
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\eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.}
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}
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}
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Invalid arguments will result in return value \code{NaN}, with a warning.
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Invalid arguments will result in return value \code{NaN}, with a warning.
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The length of the result is determined by \code{n} for
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The length of the result is determined by \code{n} for
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\code{rhyper}, and is the maximum of the lengths of the
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\code{rhyper}, and is the maximum of the lengths of the
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numerical arguments for the other functions.
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numerical arguments for the other functions.
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The numerical arguments other than \code{n} are recycled to the
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The numerical arguments other than \code{n} are recycled to the
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length of the result. Only the first elements of the logical
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length of the result. Only the first elements of the logical
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arguments are used.
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arguments are used.
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}
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}
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\details{
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\details{
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\deqn{
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\deqn{
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p(x) = \left. {m \choose x}{n \choose k-x} \right/ {m+n \choose k}%
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p(x) = \left. {m \choose x}{n \choose k-x} \right/ {m+n \choose k}%
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}{p(x) = choose(m, x) choose(n, k-x) / choose(m+n, k)}
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}{p(x) = choose(m, x) choose(n, k-x) / choose(m+n, k)}
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for \eqn{x = 0, \ldots, k}.
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for \eqn{x = 0, \ldots, k}.
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Note that \eqn{p(x)} is non-zero only for
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\eqn{\max(0, k-n) \le x \le \min(k, m)}{max(0, k-n) <= x <= min(k, m)}.
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With \eqn{p := m/(m+n)} (hence \eqn{Np = N \times p} in the
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reference's notation), the first two moments are mean
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\deqn{E[X] = \mu = k p} and variance
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\deqn{\mbox{Var}(X) = k p (1 - p) \frac{m+n-k}{m+n-1},}{%
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Var(X) = k p (1 - p) * (m+n-k)/(m+n-1),}
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which shows the closeness to the Binomial\eqn{(k,p)} (where the
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hypergeometric has smaller variance unless \eqn{k = 1}).
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The quantile is defined as the smallest value \eqn{x} such that
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The quantile is defined as the smallest value \eqn{x} such that
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\eqn{F(x) \ge p}, where \eqn{F} is the distribution function.
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\eqn{F(x) \ge p}, where \eqn{F} is the distribution function.
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If one of \eqn{m, n, k}, exceeds \code{\link{.Machine}$integer.max},
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currently the equivalent of \code{qhyper(runif(nn), m,n,k)} is used,
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when a binomial approximation may be considerably more efficient.
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}
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}
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\source{
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\source{
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\code{dhyper} computes via binomial probabilities, using code
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\code{dhyper} computes via binomial probabilities, using code
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contributed by Catherine Loader (see \code{\link{dbinom}}).
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contributed by Catherine Loader (see \code{\link{dbinom}}).
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