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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.cauchy_dist"></a><a class="link" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy-Lorentz
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Distribution</a>
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</h4></div></div></div>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cauchy_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="identifier">cauchy</span><span class="special">;</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cauchy_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
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<span class="identifier">cauchy_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
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<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz
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distribution</a> is named after Augustin Cauchy and Hendrik Lorentz.
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It is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous
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probability distribution</a> with <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
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distribution function PDF</a> given by:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../equations/cauchy_ref1.svg"></span>
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</p>
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<p>
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The location parameter x<sub>0</sub>   is the location of the peak of the distribution
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(the mode of the distribution), while the scale parameter γ   specifies half
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the width of the PDF at half the maximum height. If the location is zero,
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and the scale 1, then the result is a standard Cauchy distribution.
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</p>
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<p>
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The distribution is important in physics as it is the solution to the differential
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equation describing forced resonance, while in spectroscopy it is the description
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of the line shape of spectral lines.
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</p>
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<p>
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The following graph shows how the distributions moves as the location parameter
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changes:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf1.svg" align="middle"></span>
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</p>
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<p>
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While the following graph shows how the shape (scale) parameter alters
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the distribution:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf2.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.cauchy_dist.h0"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.member_functions"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist_ref.dists.cauchy_dist.member_functions">Member
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Functions</a>
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</h5>
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<pre class="programlisting"><span class="identifier">cauchy_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
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</pre>
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<p>
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Constructs a Cauchy distribution, with location parameter <span class="emphasis"><em>location</em></span>
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and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
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their default values (location = 0, scale = 1) then the result is a Standard
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Cauchy Distribution.
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</p>
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<p>
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Requires scale > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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</pre>
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<p>
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Returns the location parameter of the distribution.
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</p>
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<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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</pre>
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<p>
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Returns the scale parameter of the distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.cauchy_dist.h1"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.non_member_accessors"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist_ref.dists.cauchy_dist.non_member_accessors">Non-member
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Accessors</a>
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</h5>
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<p>
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All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
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functions</a> that are generic to all distributions are supported:
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
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</p>
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<p>
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Note however that the Cauchy distribution does not have a mean, standard
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deviation, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
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undefined function</a> to control whether these should fail to compile
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with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
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</p>
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<p>
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Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>
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and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
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will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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if called.
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</p>
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<p>
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The domain of the random variable is [-[max_value], +[min_value]].
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.cauchy_dist.h2"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.accuracy"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist_ref.dists.cauchy_dist.accuracy">Accuracy</a>
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</h5>
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<p>
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The Cauchy distribution is implemented in terms of the standard library
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<code class="computeroutput"><span class="identifier">tan</span></code> and <code class="computeroutput"><span class="identifier">atan</span></code>
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functions, and as such should have very low error rates.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.cauchy_dist.h3"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.implementation"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist_ref.dists.cauchy_dist.implementation">Implementation</a>
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</h5>
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<p>
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In the following table x<sub>0 </sub> is the location parameter of the distribution,
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γ   is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate,
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<span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Implementation Notes
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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pdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: pdf = 1 / (π * γ * (1 + ((x - x<sub>0 </sub>) / γ)<sup>2</sup>)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf and its complement
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</p>
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</td>
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<td>
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<p>
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The cdf is normally given by:
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</p>
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<p>
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p = 0.5 + atan(x)/π
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</p>
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<p>
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But that suffers from cancellation error as x -> -∞. So recall
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that for <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span>
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<span class="number">0</span></code>:
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</p>
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<p>
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atan(x) = -π/2 - atan(1/x)
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</p>
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<p>
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Substituting into the above we get:
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</p>
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<p>
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p = -atan(1/x) ; x < 0
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</p>
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<p>
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So the procedure is to calculate the cdf for -fabs(x) using the
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above formula. Note that to factor in the location and scale
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parameters you must substitute (x - x<sub>0 </sub>) / γ   for x in the above.
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</p>
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<p>
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This procedure yields the smaller of <span class="emphasis"><em>p</em></span> and
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<span class="emphasis"><em>q</em></span>, so the result may need subtracting from
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1 depending on whether we want the complement or not, and whether
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<span class="emphasis"><em>x</em></span> is less than x<sub>0 </sub> or not.
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile
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</p>
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</td>
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<td>
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<p>
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The same procedure is used irrespective of whether we're starting
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from the probability or its complement. First the argument <span class="emphasis"><em>p</em></span>
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is reduced to the range [-0.5, 0.5], then the relation
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</p>
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<p>
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x = x<sub>0 </sub> ± γ   / tan(π * p)
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</p>
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<p>
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is used to obtain the result. Whether we're adding or subtracting
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from x<sub>0 </sub> is determined by whether we're starting from the complement
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or not.
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mode
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</p>
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</td>
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<td>
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<p>
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The location parameter.
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.cauchy_dist.h4"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.references"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist_ref.dists.cauchy_dist.references">References</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz
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distribution</a>
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</li>
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<li class="listitem">
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<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm" target="_top">NIST
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Exploratory Data Analysis</a>
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</li>
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<li class="listitem">
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<a href="http://mathworld.wolfram.com/CauchyDistribution.html" target="_top">Weisstein,
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Eric W. "Cauchy Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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