Cauchy
The Cauchy Distribution
Density, distribution function, quantile function and random
generation for the Cauchy distribution with location parameter
location and scale parameter scale.
- Keywords
- distribution
Usage
dcauchy(x, location = 0, scale = 1, log = FALSE)
pcauchy(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qcauchy(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rcauchy(n, location = 0, scale = 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. - location, scale
- location and scale parameters.
- 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 location or scale are not specified, they assume
the default values of 0 and 1 respectively.
The Cauchy distribution with location $l$ and scale $s$ has density $$f(x) = \frac{1}{\pi s} \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}% $$ for all $x$.
Value
dcauchy, pcauchy, and qcauchy are respectively
the density, distribution function and quantile function of the Cauchy
distribution. rcauchy generates random deviates from the
Cauchy.The length of the result is determined by n for
rcauchy, 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.
Source
dcauchy, pcauchy and qcauchy are all calculated
from numerically stable versions of the definitions. rcauchy uses inversion.
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 16. Wiley, New York.
See Also
Distributions for other standard distributions, including
dt for the t distribution which generalizes
dcauchy(*, l = 0, s = 1).
Examples
library(stats)
dcauchy(-1:4)