The Cauchy Distribution

Description

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

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

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 gives the density, pcauchy is the cumulative distribution function, and qcauchy is the quantile function of the Cauchy distribution. rcauchy generates random deviates.

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

⁠Johnson NL, Kotz S, Balakrishnan N (1994). Continuous Univariate Distributions, volume 1. Wiley, New York. ISBN 978-0-471-58495-7.
Chapter 16.

See Also

Distributions for other standard distributions, including dt for the t distribution which generalizes dcauchy(*, l = 0, s = 1).

Examples

dcauchy(-1:4)