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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.rayleigh"></a><a class="link" href="rayleigh.html" title="Rayleigh Distribution">Rayleigh 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">rayleigh</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">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>
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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">double</span><span class="special">,</span>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 20. 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">rayleigh_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">rayleigh_distribution</span><span class="special"><></span> <span class="identifier">rayleigh</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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">rayleigh_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="comment">// Construct:</span>
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<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>
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<span class="comment">// Accessors:</span>
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<span class="identifier">RealType</span> <span class="identifier">sigma</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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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
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distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
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density function</a>:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">f(x; sigma) = x * exp(-x<sup>2</sup>/2 σ<sup>2</sup>) / σ<sup>2</sup></span>
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</p></blockquote></div>
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<p>
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For sigma parameter <span class="emphasis"><em>σ</em></span> > 0, and <span class="emphasis"><em>x</em></span>
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> 0.
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</p>
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<p>
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The Rayleigh distribution is often used where two orthogonal components
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have an absolute value, for example, wind velocity and direction may be
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combined to yield a wind speed, or real and imaginary components may have
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absolute values that are Rayleigh distributed.
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</p>
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<p>
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The following graph illustrates how the Probability density Function(pdf)
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varies with the shape parameter σ:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_pdf.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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and the Cumulative Distribution Function (cdf)
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_cdf.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.rayleigh.h0"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.related_distributions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.related_distributions">Related
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distributions</a>
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</h5>
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<p>
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The absolute value of two independent normal distributions X and Y, √ (X<sup>2</sup> +
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Y<sup>2</sup>) is a Rayleigh distribution.
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</p>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Chi_distribution" target="_top">Chi</a>,
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<a href="http://en.wikipedia.org/wiki/Rice_distribution" target="_top">Rice</a>
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and <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull</a>
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distributions are generalizations of the <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
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distribution</a>.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.rayleigh.h1"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.member_functions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.member_functions">Member
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Functions</a>
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</h5>
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<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>
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</pre>
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<p>
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Constructs a <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
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distribution</a> with σ <span class="emphasis"><em>sigma</em></span>.
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</p>
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<p>
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Requires that the σ parameter is greater than zero, 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">sigma</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 <span class="emphasis"><em>sigma</em></span> parameter of this distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.rayleigh.h2"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.non_member_accessors"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.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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The domain of the random variable is [0, max_value].
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.rayleigh.h3"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.accuracy"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.accuracy">Accuracy</a>
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</h5>
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<p>
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The Rayleigh distribution is implemented in terms of the standard library
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<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
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error rates. Some constants such as skewness and kurtosis were calculated
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using NTL RR type with 150-bit accuracy, about 50 decimal digits.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.rayleigh.h4"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.implementation"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.implementation">Implementation</a>
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</h5>
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<p>
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In the following table σ is the sigma parameter of the distribution, <span class="emphasis"><em>x</em></span>
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is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
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= 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 = x * exp(-x<sup>2</sup>)/2 σ<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
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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: p = 1 - exp(-x<sup>2</sup>/2) σ<sup>2</sup>= -<a class="link" href="../../powers/expm1.html" title="expm1">expm1</a>(-x<sup>2</sup>/2)
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σ<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 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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Using the relation: q = exp(-x<sup>2</sup>/ 2) * σ<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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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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Using the relation: x = sqrt(-2 * σ <sup>2</sup>) * log(1 - p)) = sqrt(-2
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* σ <sup>2</sup>) * <a class="link" href="../../powers/log1p.html" title="log1p">log1p</a>(-p))
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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 from the 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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Using the relation: x = sqrt(-2 * σ <sup>2</sup>) * log(q))
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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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mean
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</p>
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</td>
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<td>
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<p>
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σ * sqrt(π/2)
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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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variance
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</p>
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</td>
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<td>
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<p>
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σ<sup>2</sup> * (4 - π/2)
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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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σ
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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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skewness
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</p>
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</td>
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<td>
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<p>
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Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
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Eric W. "Weibull Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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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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kurtosis
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</p>
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</td>
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<td>
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<p>
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Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
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Eric W. "Weibull Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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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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kurtosis excess
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</p>
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</td>
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<td>
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<p>
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Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
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Eric W. "Weibull Distribution." From MathWorld--A Wolfram
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Web Resource.</a>
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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.rayleigh.h5"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.references"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.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/Rayleigh_distribution" target="_top">http://en.wikipedia.org/wiki/Rayleigh_distribution</a>
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</li>
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<li class="listitem">
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<a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
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Eric W. "Rayleigh 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 © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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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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<div class="spirit-nav">
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<a accesskey="p" href="poisson_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="skew_normal_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
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