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% File src/library/stats/man/Poisson.Rd
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% File src/library/stats/man/Poisson.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{Poisson}
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\name{Poisson}
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\alias{Poisson}
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\alias{Poisson}
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\alias{dpois}
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\alias{dpois}
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  Invalid \code{lambda} will result in return value \code{NaN}, with a warning.
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  Invalid \code{lambda} 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{rpois}, and is the maximum of the lengths of the
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  \code{rpois}, 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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%-- please leave the linebreaking for the next two ! --
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%-- please leave the linebreaking for the next two ! --
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  \deqn{p(x) = \frac{\lambda^x e^{-\lambda}}{x!}}{p(x) = \lambda^x exp(-\lambda)/x!}
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  \deqn{p(x) = \frac{\lambda^x e^{-\lambda}}{x!}}{p(x) = \lambda^x exp(-\lambda)/x!}
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  for \eqn{x = 0, 1, 2, \ldots} .
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  for \eqn{x = 0, 1, 2, \ldots} .
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  The mean and variance are \eqn{E(X) = Var(X) = \lambda}.
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  The mean and variance are \eqn{E(X) = Var(X) = \lambda}.
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  Note that \eqn{\lambda = 0} is really a limit case (setting
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  \eqn{0^0 = 1}) resulting in a point mass at \eqn{0}, see also the example.
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  If an element of \code{x} is not integer, the result of \code{dpois}
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  If an element of \code{x} is not integer, the result of \code{dpois}
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  is zero, with a warning.
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  is zero, with a warning.
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  \eqn{p(x)} is computed using Loader's algorithm, see the reference in
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  \eqn{p(x)} is computed using Loader's algorithm, see the reference in
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  \code{\link{dbinom}}.
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  \code{\link{dbinom}}.
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par(mfrow = c(2, 1))
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par(mfrow = c(2, 1))
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x <- seq(-0.01, 5, 0.01)
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x <- seq(-0.01, 5, 0.01)
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plot(x, ppois(x, 1), type = "s", ylab = "F(x)", main = "Poisson(1) CDF")
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plot(x, ppois(x, 1), type = "s", ylab = "F(x)", main = "Poisson(1) CDF")
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plot(x, pbinom(x, 100, 0.01), type = "s", ylab = "F(x)",
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plot(x, pbinom(x, 100, 0.01), type = "s", ylab = "F(x)",
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     main = "Binomial(100, 0.01) CDF")
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     main = "Binomial(100, 0.01) CDF")
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## The (limit) case  lambda = 0 :
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stopifnot(identical(dpois(0,0), 1),
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	  identical(ppois(0,0), 1),
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	  identical(qpois(1,0), 0))
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}
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}
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\keyword{distribution}
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\keyword{distribution}