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

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) = \frac{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.

Value

dnorm gives the density, pnorm is the cumulative distribution function, and qnorm is the quantile function of the normal distribution. rnorm generates random deviates.

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

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

For sd = 0 this gives the limit as sd decreases to 0, a point mass at mu. sd < 0 is an error and returns NaN.

Source

For pnorm, based on ⁠Cody (1993).

For qnorm, the code is based on a C translation of ⁠Wichura (1988) which provides precise results up to about 16 digits for 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 ⁠Mächler (2022).

For rnorm, see RNG for how to select the algorithm and for references to the supplied methods.

References

⁠Becker RA, Chambers JM, Wilks AR (1988). The New S Language. Chapman and Hall/CRC, London.

⁠Cody WJ (1993). “Algorithm 715: SPECFUN—A Portable FORTRAN Package of Special Function Routines and Test Drivers.” ACM Transactions on Mathematical Software, 19(1), 22–30. doi:10.1145/151271.151273.

⁠Johnson NL, Kotz S, Balakrishnan N (1994). Continuous Univariate Distributions, volume 1. Wiley, New York. ISBN 978-0-471-58495-7.
Chapter 13.

⁠Mächler M (2022). Asymptotic Tail Formulas for Gaussian Quantiles. Vignette, CRAN package DPQ, https://CRAN.R-project.org/package=DPQ/vignettes/qnorm-asymp.pdf.

⁠Wichura MJ (1988). “Algorithm AS 241: The Percentage Points of the Normal Distribution.” Applied Statistics, 37(3), 477. doi:10.2307/2347330.

See Also

Distributions for other standard distributions, including 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)