Normal
The Normal Distribution
Density, distribution function, quantile function and random
generation for the normal distribution with mean equal to mean
and standard deviation equal to sd.
- Keywords
- distribution
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 \(P[X \le x]\) 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 $$ f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/2\sigma^2}$$ where \(\mu\) is the mean of the distribution and \(\sigma\) the standard deviation.
Value
dnorm gives the density,
pnorm gives the distribution function,
qnorm gives the quantile function, and
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.
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
Distributions for other standard distributions, including
dlnorm for the Lognormal distribution.
Examples
library(stats)
# NOT RUN {
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)
# }