\HeaderA{Geometric}{The Geometric Distribution}{Geometric}
\aliasA{dgeom}{Geometric}{dgeom}
\aliasA{pgeom}{Geometric}{pgeom}
\aliasA{qgeom}{Geometric}{qgeom}
\aliasA{rgeom}{Geometric}{rgeom}
\keyword{distribution}{Geometric}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the geometric distribution with parameter \code{prob}.
\end{Description}
\begin{Usage}
\begin{verbatim}
dgeom(x, prob, log = FALSE)
pgeom(q, prob, lower.tail = TRUE, log.p = FALSE)
qgeom(p, prob, lower.tail = TRUE, log.p = FALSE)
rgeom(n, prob)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, q}] vector of quantiles representing the number of failures in
a sequence of Bernoulli trials before success occurs.
\item[\code{p}] vector of probabilities.
\item[\code{n}] number of observations. If \code{length(n) > 1}, the length
is taken to be the number required.
\item[\code{prob}] probability of success in each trial. \code{0 < prob <= 1}.
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The geometric distribution with \code{prob} \eqn{= p}{} has density
\deqn{p(x) = p {(1-p)}^{x}}{p(x) = p (1-p)^x}
for \eqn{x = 0, 1, 2, \ldots}{x = 0, 1, 2, ...}, \eqn{0 < p \le 1}{}.

If an element of \code{x} is not integer, the result of \code{pgeom}
is zero, with a warning.

The quantile is defined as the smallest value \eqn{x}{} such that
\eqn{F(x) \ge p}{}, where \eqn{F}{} is the distribution function.
\end{Details}
\begin{Value}
\code{dgeom} gives the density,
\code{pgeom} gives the distribution function,
\code{qgeom} gives the quantile function, and
\code{rgeom} generates random deviates.

Invalid \code{prob} will result in return value \code{NaN}, with a warning.
\end{Value}
\begin{Source}\relax
\code{dgeom} computes via \code{dbinom}, using code contributed by
Catherine Loader (see \code{\LinkA{dbinom}{dbinom}}).

\code{pgeom} and \code{qgeom} are based on the closed-form formulae.

\code{rgeom} uses the derivation as an exponential mixture of Poissons, see

Devroye, L. (1986) \emph{Non-Uniform Random Variate Generation.}
Springer-Verlag, New York. Page 480.
\end{Source}
\begin{SeeAlso}\relax
\code{\LinkA{dnbinom}{dnbinom}} for the negative binomial which generalizes
the geometric distribution.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
qgeom((1:9)/10, prob = .2)
Ni <- rgeom(20, prob = 1/4); table(factor(Ni, 0:max(Ni)))
\end{ExampleCode}
\end{Examples}

