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<a name="Cauchy-Random-Variable"></a>
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<p>
Next: <a href="maxima_238.html#Gumbel-Random-Variable" accesskey="n" rel="next">Gumbel Random Variable</a>, Previous: <a href="maxima_236.html#Laplace-Random-Variable" accesskey="p" rel="previous">Laplace Random Variable</a>, Up: <a href="maxima_220.html#Functions-and-Variables-for-continuous-distributions" accesskey="u" rel="up">Functions and Variables for continuous distributions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Cauchy-Random-Variable-1"></a>
<h4 class="subsection">52.2.17 Cauchy Random Variable</h4>
<p>The <em>Cauchy</em> distribution (also known as the Lorentz
distribution) is the distribution of of the ratio of two independent
normally distributed random variables with mean zero.
</p>
<p>Note that the mean, variance, standard deviation, skewness
coefficient, and kurtosis coefficient are all undefined for the Cauchy
distribution. The integrals do not converge in this case.
</p>
<a name="pdf_005fcauchy"></a><a name="Item_003a-distrib_002fdeffn_002fpdf_005fcauchy"></a><dl>
<dt><a name="index-pdf_005fcauchy"></a>Function: <strong>pdf_cauchy</strong> <em>(<var>x</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the density function of a
\({\it Cauchy}(a,b)\) random variable, with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The pdf is
$$
f(x; a, b) = {b\over \pi\left((x-a)^2+b^2\right)}
$$</p>
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<a name="cdf_005fcauchy"></a><a name="Item_003a-distrib_002fdeffn_002fcdf_005fcauchy"></a><dl>
<dt><a name="index-cdf_005fcauchy"></a>Function: <strong>cdf_cauchy</strong> <em>(<var>x</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the value at <var>x</var> of the distribution function of a
\({\it Cauchy}(a,b)\) random variable, with <em>b>0</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
<p>The cdf is
$$
F(x; a, b) = {1\over 2} + {1\over \pi} \tan^{-1} {x-a\over b}
$$</p>
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</dd></dl>
<a name="quantile_005fcauchy"></a><a name="Item_003a-distrib_002fdeffn_002fquantile_005fcauchy"></a><dl>
<dt><a name="index-quantile_005fcauchy"></a>Function: <strong>quantile_cauchy</strong> <em>(<var>q</var>,<var>a</var>,<var>b</var>)</em></dt>
<dd><p>Returns the <var>q</var>-quantile of a
\({\it Cauchy}(a,b)\) random variable, with <em>b>0</em>; in other words, this is the inverse of <code>cdf_cauchy</code>. Argument <var>q</var> must be an element of <em>[0,1]</em>. To make use of this function, write first <code>load("distrib")</code>.
</p>
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<a name="random_005fcauchy"></a><a name="Item_003a-distrib_002fdeffn_002frandom_005fcauchy"></a><dl>
<dt><a name="index-random_005fcauchy"></a>Function: <strong>random_cauchy</strong> <em>(<var>a</var>,<var>b</var>) <br> <tt>random_cauchy</tt> (<var>a</var>,<var>b</var>,<var>n</var>)</em></dt>
<dd>
<p>Returns a
\({\it Cauchy}(a,b)\) random variate, with <em>b>0</em>. Calling <code>random_cauchy</code> with a third argument <var>n</var>, a random sample of size <var>n</var> will be simulated.
</p>
<p>The implemented algorithm is based on the general inverse method.
</p>
<p>To make use of this function, write first <code>load("distrib")</code>.
</p>
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Next: <a href="maxima_238.html#Gumbel-Random-Variable" accesskey="n" rel="next">Gumbel Random Variable</a>, Previous: <a href="maxima_236.html#Laplace-Random-Variable" accesskey="p" rel="previous">Laplace Random Variable</a>, Up: <a href="maxima_220.html#Functions-and-Variables-for-continuous-distributions" accesskey="u" rel="up">Functions and Variables for continuous distributions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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