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<title>Rayleigh Distribution</title>
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<div class="section math_toolkit_dist_dist_ref_dists_rayleigh">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.dist.dist_ref.dists.rayleigh"></a><a class="link" href="rayleigh.html" title="Rayleigh Distribution">Rayleigh
          Distribution</a>
</h5></div></div></div>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">rayleigh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<p>
          </p>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a>   <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">rayleigh_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">rayleigh_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">rayleigh</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</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="Policies">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">rayleigh_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
   <span class="comment">// Construct:</span>
   <span class="identifier">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">1</span><span class="special">)</span>
   <span class="comment">// Accessors:</span>
   <span class="identifier">RealType</span> <span class="identifier">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
            The <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
            distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
            density function</a>:
          </p>
<p>
            f(x; sigma) = x * exp(-x<sup>2</sup>/2 &#963;<sup>2</sup>) / &#963;<sup>2</sup>
          </p>
<p>
            For sigma parameter &#963; &#160; &gt; 0, and x &gt; 0.
          </p>
<p>
            The Rayleigh distribution is often used where two orthogonal components
            have an absolute value, for example, wind velocity and direction may
            be combined to yield a wind speed, or real and imaginary components may
            have absolute values that are Rayleigh distributed.
          </p>
<p>
            The following graph illustrates how the Probability density Function(pdf)
            varies with the shape parameter &#963;:
          </p>
<p>
            <span class="inlinemediaobject"><img src="../../../../../graphs/rayleigh_pdf.png" align="middle"></span>
          </p>
<p>
            and the Cumulative Distribution Function (cdf)
          </p>
<p>
            <span class="inlinemediaobject"><img src="../../../../../graphs/rayleigh_cdf.png" align="middle"></span>
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.rayleigh.h0"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.rayleigh.related_distributions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist.dist_ref.dists.rayleigh.related_distributions">Related
            distributions</a>
          </h5>
<p>
            The absolute value of two independent normal distributions X and Y, &#8730; (X<sup>2</sup> +
            Y<sup>2</sup>) is a Rayleigh distribution.
          </p>
<p>
            The <a href="http://en.wikipedia.org/wiki/Chi_distribution" target="_top">Chi</a>,
            <a href="http://en.wikipedia.org/wiki/Rice_distribution" target="_top">Rice</a>
            and <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull</a>
            distributions are generalizations of the <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
            distribution</a>.
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.rayleigh.h1"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.rayleigh.member_functions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist.dist_ref.dists.rayleigh.member_functions">Member
            Functions</a>
          </h5>
<pre class="programlisting"><span class="identifier">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
            Constructs a <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
            distribution</a> with &#963; <span class="emphasis"><em>sigma</em></span>.
          </p>
<p>
            Requires that the &#963; parameter is greater than zero, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
          </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
            Returns the <span class="emphasis"><em>sigma</em></span> parameter of this distribution.
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.rayleigh.h2"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.rayleigh.non_member_accessors"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist.dist_ref.dists.rayleigh.non_member_accessors">Non-member
            Accessors</a>
          </h5>
<p>
            All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
            accessor functions</a> that are generic to all distributions are supported:
            <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>,
            <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard
            Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>,
            <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>,
            <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>,
            <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>,
            <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>,
            <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>.
          </p>
<p>
            The domain of the random variable is [0, max_value].
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.rayleigh.h3"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.rayleigh.accuracy"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist.dist_ref.dists.rayleigh.accuracy">Accuracy</a>
          </h5>
<p>
            The Rayleigh distribution is implemented in terms of the standard library
            <code class="computeroutput"><span class="identifier">sqrt</span></code> and <code class="computeroutput"><span class="identifier">exp</span></code> and as such should have very low
            error rates. Some constants such as skewness and kurtosis were calculated
            using NTL RR type with 150-bit accuracy, about 50 decimal digits.
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.rayleigh.h4"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.rayleigh.implementation"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist.dist_ref.dists.rayleigh.implementation">Implementation</a>
          </h5>
<p>
            In the following table &#963; &#160; is the sigma parameter of the distribution, <span class="emphasis"><em>x</em></span>
            is the random variate, <span class="emphasis"><em>p</em></span> is the probability and
            <span class="emphasis"><em>q = 1-p</em></span>.
          </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                    <p>
                      Function
                    </p>
                  </th>
<th>
                    <p>
                      Implementation Notes
                    </p>
                  </th>
</tr></thead>
<tbody>
<tr>
<td>
                    <p>
                      pdf
                    </p>
                  </td>
<td>
                    <p>
                      Using the relation: pdf = x * exp(-x<sup>2</sup>)/2 &#963;<sup>2</sup>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      cdf
                    </p>
                  </td>
<td>
                    <p>
                      Using the relation: p = 1 - exp(-x<sup>2</sup>/2) &#963;<sup>2</sup> &#160; = -<a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a>(-x<sup>2</sup>/2)
                      &#963;<sup>2</sup>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      cdf complement
                    </p>
                  </td>
<td>
                    <p>
                      Using the relation: q = exp(-x<sup>2</sup>/ 2) * &#963;<sup>2</sup>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      quantile
                    </p>
                  </td>
<td>
                    <p>
                      Using the relation: x = sqrt(-2 * &#963; <sup>2</sup>) * log(1 - p)) = sqrt(-2
                      * &#963; <sup>2</sup>) * <a class="link" href="../../../special/powers/log1p.html" title="log1p">log1p</a>(-p))
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      quantile from the complement
                    </p>
                  </td>
<td>
                    <p>
                      Using the relation: x = sqrt(-2 * &#963; <sup>2</sup>) * log(q))
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      mean
                    </p>
                  </td>
<td>
                    <p>
                      &#963; * sqrt(&#960;/2)
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      variance
                    </p>
                  </td>
<td>
                    <p>
                      &#963;<sup>2</sup> * (4 - &#960;/2)
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      mode
                    </p>
                  </td>
<td>
                    <p>
                      &#963;
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      skewness
                    </p>
                  </td>
<td>
                    <p>
                      Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                      Eric W. "Weibull Distribution." From MathWorld--A
                      Wolfram Web Resource.</a>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      kurtosis
                    </p>
                  </td>
<td>
                    <p>
                      Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                      Eric W. "Weibull Distribution." From MathWorld--A
                      Wolfram Web Resource.</a>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      kurtosis excess
                    </p>
                  </td>
<td>
                    <p>
                      Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                      Eric W. "Weibull Distribution." From MathWorld--A
                      Wolfram Web Resource.</a>
                    </p>
                  </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.rayleigh.h5"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.rayleigh.references"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist.dist_ref.dists.rayleigh.references">References</a>
          </h5>
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
<li class="listitem">
                <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">http://en.wikipedia.org/wiki/Rayleigh_distribution</a>
              </li>
<li class="listitem">
                <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
                Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram
                Web Resource.</a>
              </li>
</ul></div>
</div>
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<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
      Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        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>)
      </p>
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