Cauchy {stats}R Documentation

The Cauchy Distribution


Density, distribution function, quantile function and random generation for the Cauchy distribution with location parameter location and scale parameter scale.


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)


x, q

vector of quantiles.


vector of probabilities.


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).


logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].


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) = 1 / (π s (1 + ((x-l)/s)^2))

for all x.


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.


dcauchy, pcauchy and qcauchy are all calculated from numerically stable versions of the definitions.

rcauchy uses inversion.


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).



[Package stats version 3.3.0 Index]