Normal                 package:stats                 R Documentation

The Normal Distribution

Description:

     Density, distribution function, quantile function and random
     generation for the normal distribution with mean equal to 'mean'
     and standard deviation equal to '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:

     x,q: vector of quantiles.

       p: vector of probabilities.

       n: number of observations. If 'length(n) > 1', the length is
          taken to be the number required.

    mean: vector of means.

      sd: vector of standard deviations.

log, log.p: logical; if TRUE, probabilities p are given as log(p).

lower.tail: logical; if TRUE (default), probabilities are P[X <= x],
          otherwise, P[X > x].

Details:

     If 'mean' or 'sd' are not specified they assume the default values
     of '0' and '1', respectively.

     The normal distribution has density

       f(x) = 1/(sqrt(2 pi) sigma) e^-((x - mu)^2/(2 sigma^2))

     where mu is the mean of the distribution and sigma the standard
     deviation.

     'qnorm' is based on Wichura's algorithm AS 241 which provides
     precise results up to about 16 digits.

Value:

     'dnorm' gives the density, 'pnorm' gives the distribution
     function, 'qnorm' gives the quantile function, and 'rnorm'
     generates random deviates.

Source:

     For 'pnorm', based on

     Cody, W. D. (1993) Algorithm 715: SPECFUN - A portable FORTRAN
     package of special function routines and test drivers. _ACM
     Transactions on Mathematical Software_ *19*, 22-32.

     For 'qnorm', the code is a C translation of

     Wichura, M. J. (1988) Algorithm AS 241: The Percentage Points of
     the Normal Distribution. _Applied Statistics_, *37*, 477-484.

     For 'rnorm', see 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) _The New S
     Language_. Wadsworth & Brooks/Cole.

     Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) _Continuous
     Univariate Distributions_, volume 1, chapter 13. Wiley, New York.

See Also:

     'runif' and '.Random.seed' about random number generation, and
     'dlnorm' for the _Log_normal distribution.

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)

