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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"><html xmlns="http://www.w3.org/1999/xhtml"><head><title>R: The Cauchy Distribution</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link rel="stylesheet" type="text/css" href="R.css" /> </head><body> <table width="100%" summary="page for Cauchy {stats}"><tr><td>Cauchy {stats}</td><td style="text-align: right;">R Documentation</td></tr></table> <h2>The Cauchy Distribution</h2> <h3>Description</h3> <p>Density, distribution function, quantile function and random generation for the Cauchy distribution with location parameter <code>location</code> and scale parameter <code>scale</code>. </p> <h3>Usage</h3> <pre> 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) </pre> <h3>Arguments</h3> <table summary="R argblock"> <tr valign="top"><td><code>x, q</code></td> <td> <p>vector of quantiles.</p> </td></tr> <tr valign="top"><td><code>p</code></td> <td> <p>vector of probabilities.</p> </td></tr> <tr valign="top"><td><code>n</code></td> <td> <p>number of observations. If <code>length(n) > 1</code>, the length is taken to be the number required.</p> </td></tr> <tr valign="top"><td><code>location, scale</code></td> <td> <p>location and scale parameters.</p> </td></tr> <tr valign="top"><td><code>log, log.p</code></td> <td> <p>logical; if TRUE, probabilities p are given as log(p).</p> </td></tr> <tr valign="top"><td><code>lower.tail</code></td> <td> <p>logical; if TRUE (default), probabilities are <i>P[X ≤ x]</i>, otherwise, <i>P[X > x]</i>.</p> </td></tr> </table> <h3>Details</h3> <p>If <code>location</code> or <code>scale</code> are not specified, they assume the default values of <code>0</code> and <code>1</code> respectively. </p> <p>The Cauchy distribution with location <i>l</i> and scale <i>s</i> has density </p> <p style="text-align: center;"><i>f(x) = 1 / (π s (1 + ((x-l)/s)^2))</i></p> <p>for all <i>x</i>. </p> <h3>Value</h3> <p><code>dcauchy</code>, <code>pcauchy</code>, and <code>qcauchy</code> are respectively the density, distribution function and quantile function of the Cauchy distribution. <code>rcauchy</code> generates random deviates from the Cauchy. </p> <p>The length of the result is determined by <code>n</code> for <code>rcauchy</code>, and is the maximum of the lengths of the numerical arguments for the other functions. </p> <p>The numerical arguments other than <code>n</code> are recycled to the length of the result. Only the first elements of the logical arguments are used. </p> <h3>Source</h3> <p><code>dcauchy</code>, <code>pcauchy</code> and <code>qcauchy</code> are all calculated from numerically stable versions of the definitions. </p> <p><code>rcauchy</code> uses inversion. </p> <h3>References</h3> <p>Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) <em>The New S Language</em>. Wadsworth & Brooks/Cole. </p> <p>Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) <em>Continuous Univariate Distributions</em>, volume 1, chapter 16. Wiley, New York. </p> <h3>See Also</h3> <p><a href="Distributions.html">Distributions</a> for other standard distributions, including <code><a href="TDist.html">dt</a></code> for the t distribution which generalizes <code>dcauchy(*, l = 0, s = 1)</code>. </p> <h3>Examples</h3> <pre> dcauchy(-1:4) </pre> <hr /><div style="text-align: center;">[Package <em>stats</em> version 3.6.0 <a href="00Index.html">Index</a>]</div> </body></html>