Normal                 package:base                 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.

References:

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

See Also:

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

Examples:

     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=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)

