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% File src/library/stats/man/Normal.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2023 R Core Team
% Distributed under GPL 2 or later

\name{Normal}
\alias{Normal}
\alias{dnorm}
\alias{pnorm}
\alias{qnorm}
\alias{rnorm}
% These concepts are for the last example
\concept{error function}
\concept{erf}
\concept{erfc}
\concept{erfinv}
\concept{erfcinv}
\title{The Normal Distribution}
\description{
  Density, distribution function, quantile function and random
  generation for the normal distribution with mean equal to \code{mean}
  and standard deviation equal to \code{sd}.
}
\usage{
dnorm(x, mean = 0, sd = 1, log = FALSE)
pnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
rnorm(n, mean = 0, sd = 1)
}
\arguments{
  \item{x, q}{vector of quantiles.}
  \item{p}{vector of probabilities.}
  \item{n}{number of observations. If \code{length(n) > 1}, the length
    is taken to be the number required.}
  \item{mean}{vector of means.}
  \item{sd}{vector of standard deviations.}
  \item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).}
  \item{lower.tail}{logical; if TRUE (default), probabilities are
    \eqn{P[X \le x]} otherwise, \eqn{P[X > x]}.}
}
\value{
  \code{dnorm} gives the density,
  \code{pnorm} gives the distribution function,
  \code{qnorm} gives the quantile function, and
  \code{rnorm} generates random deviates.

  The length of the result is determined by \code{n} for
  \code{rnorm}, and is the maximum of the lengths of the
  numerical arguments for the other functions.

  The numerical arguments other than \code{n} are recycled to the
  length of the result.  Only the first elements of the logical
  arguments are used.

  For \code{sd = 0} this gives the limit as \code{sd} decreases to 0, a
  point mass at \code{mu}.
  \code{sd < 0} is an error and returns \code{NaN}.
}
\details{
  If \code{mean} or \code{sd} are not specified they assume the default
  values of \code{0} and \code{1}, respectively.

  The normal distribution has density
  \deqn{
    f(x) =
    \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/2\sigma^2}}{
    f(x) = 1/(\sqrt(2 \pi) \sigma) e^-((x - \mu)^2/(2 \sigma^2))
  }
  where \eqn{\mu} is the mean of the distribution and
  \eqn{\sigma} the standard deviation.
}
\seealso{
  \link{Distributions} for other standard distributions, including
  \code{\link{dlnorm}} for the \emph{Log}normal distribution.
}
\source{
  For \code{pnorm}, based on

  Cody, W. D. (1993)
  Algorithm 715: SPECFUN -- A portable FORTRAN package of special
  function routines and test drivers.
  \emph{ACM Transactions on Mathematical Software} \bold{19}, 22--32.

  For \code{qnorm}, the code is based on a C translation of

  Wichura, M. J. (1988)
  Algorithm AS 241: The percentage points of the normal distribution.
  \emph{Applied Statistics}, \bold{37}, 477--484; \doi{10.2307/2347330}.

  which provides precise results up to about 16 digits for
  \code{log.p=FALSE}.  For log scale probabilities in the extreme tails,
  since \R version 4.1.0, extensively since 4.3.0, asymptotic expansions
  are used which have been derived and explored in

  Maechler, M. (2022)
  Asymptotic tail formulas for gaussian quantiles; \CRANpkg{DPQ} vignette
  \url{https://CRAN.R-project.org/package=DPQ/vignettes/qnorm-asymp.pdf}.

  For \code{rnorm}, see \link{RNG} for how to select the algorithm and
  for references to the supplied methods.
}
\references{
  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
  \emph{The New S Language}.
  Wadsworth & Brooks/Cole.

  Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
  \emph{Continuous Univariate Distributions}, volume 1, chapter 13.
  Wiley, New York.
}
\examples{
require(graphics)

dnorm(0) == 1/sqrt(2*pi)
dnorm(1) == exp(-1/2)/sqrt(2*pi)
dnorm(1) == 1/sqrt(2*pi*exp(1))

## Using "log = TRUE" for an extended range :
par(mfrow = c(2,1))
plot(function(x) dnorm(x, log = TRUE), -60, 50,
     main = "log { Normal density }")
curve(log(dnorm(x)), add = TRUE, col = "red", lwd = 2)
mtext("dnorm(x, log=TRUE)", adj = 0)
mtext("log(dnorm(x))", col = "red", adj = 1)

plot(function(x) pnorm(x, log.p = TRUE), -50, 10,
     main = "log { Normal Cumulative }")
curve(log(pnorm(x)), add = TRUE, col = "red", lwd = 2)
mtext("pnorm(x, log=TRUE)", adj = 0)
mtext("log(pnorm(x))", col = "red", adj = 1)

## if you want the so-called 'error function'
erf <- function(x) 2 * pnorm(x * sqrt(2)) - 1
## (see Abramowitz and Stegun 29.2.29)
## and the so-called 'complementary error function'
erfc <- function(x) 2 * pnorm(x * sqrt(2), lower = FALSE)
## and the inverses
erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
erfcinv <- function (x) qnorm(x/2, lower = FALSE)/sqrt(2)
}
\keyword{distribution}