[ceph.git] / ceph / src / boost / libs / math / doc / html / math_toolkit / dist_ref / nmp.html

<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Non-Member Properties</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
<link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference">
<link rel="prev" href="../dist_ref.html" title="Statistical Distributions Reference">
<link rel="next" href="dists.html" title="Distributions">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="../dist_ref.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.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="dists.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.dist_ref.nmp"></a><a class="link" href="nmp.html" title="Non-Member Properties">Non-Member Properties</a>
</h3></div></div></div>
<p>
        Properties that are common to all distributions are accessed via non-member
        getter functions: non-membership allows more of these functions to be added
        over time, as the need arises. Unfortunately the literature uses many different
        and confusing names to refer to a rather small number of actual concepts;
        refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">concept
        index</a> to find the property you want by the name you are most familiar
        with. Or use the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">function
        index</a> to go straight to the function you want if you already know
        its name.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h0"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.function_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">Function
        Index</a>
      </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
            Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
            Distribution Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
            complement of the probability</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
          </li>
</ul></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h1"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.concept_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">Conceptual
        Index</a>
      </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
            Distribution Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
            Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse Cumulative
            Distribution Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse Survival
            Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower Critical
            Value</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent Point Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability Mass Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">Q</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
            complement of the probability</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival Function</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper Critical
            Value</a>.
          </li>
<li class="listitem">
            <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
          </li>
</ul></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h2"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        The <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
        Function</a> is the probability that the variable takes a value less than
        or equal to x. It is equivalent to the integral from -infinity to x of the
        <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        For example, the following graph shows the cdf for the normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/cdf.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h3"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.ccdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
        of the Cumulative Distribution Function</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special">&lt;</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">comp</span><span class="special">);</span>
</pre>
<p>
        The complement of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a> is the probability that the variable takes a
        value greater than x. It is equivalent to the integral from x to infinity
        of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
        Function</a>, or 1 minus the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a> of x.
      </p>
<p>
        This is also known as the survival function.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        In this library, it is obtained by wrapping the arguments to the <code class="computeroutput"><span class="identifier">cdf</span></code> function in a call to <code class="computeroutput"><span class="identifier">complement</span></code>, for example:
      </p>
<pre class="programlisting"><span class="comment">// standard normal distribution object:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span> <span class="identifier">norm</span><span class="special">;</span>
<span class="comment">// print survival function for x=2.0:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">2.0</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
        For example, the following graph shows the __complement of the cdf for the
        normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/survival.png"></span>
      </p>
<p>
        See <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a> for why the complement
        is useful and when it should be used.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h4"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.hazard"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard
        Function</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">hazard</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
        of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span>
      </p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
          Some authors refer to this as the conditional failure density function
          rather than the hazard function.
        </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h5"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.chf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative
        Hazard Function</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">chf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard
        Function</a> of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span>
      </p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
          Some authors refer to this as simply the "Hazard Function".
        </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.nmp.h6"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mean"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the mean of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined mean (for example the Cauchy
        distribution).
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h7"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.median"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">median</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the median of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h8"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mode"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the mode of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined mode.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h9"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
        Density Function</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">pdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
        For a continuous function, the probability density function (pdf) returns
        the probability that the variate has the value x. Since for continuous distributions
        the probability at a single point is actually zero, the probability is better
        expressed as the integral of the pdf between two points: see the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a>.
      </p>
<p>
        For a discrete distribution, the pdf is the probability that the variate
        takes the value x.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the random variable is outside the defined range for the distribution.
      </p>
<p>
        For example, for a standard normal distribution the pdf looks like this:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/pdf.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h10"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.range"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">Range</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">range</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the valid range of the random variable over distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h11"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">p</span><span class="special">);</span>
</pre>
<p>
        The quantile is best viewed as the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a>, it returns a value <span class="emphasis"><em>x</em></span> such
        that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span>
        <span class="identifier">p</span></code>.
      </p>
<p>
        This is also known as the <span class="emphasis"><em>percent point function</em></span>, or
        <span class="emphasis"><em>percentile</em></span>, or <span class="emphasis"><em>fractile</em></span>, it is
        also the same as calculating the <span class="emphasis"><em>lower critical value</em></span>
        of a distribution.
      </p>
<p>
        This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the probability lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
        if there is no finite value that has the specified probability.
      </p>
<p>
        The following graph shows the quantile function for a standard normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/quantile.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h12"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile_c"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile
        from the complement of the probability.</a>
      </h5>
<p>
        See also <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>.
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special">&lt;</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">comp</span><span class="special">);</span>
</pre>
<p>
        This is the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
        of the Cumulative Distribution Function</a>. It is calculated by wrapping
        the arguments in a call to the quantile function in a call to <span class="emphasis"><em>complement</em></span>.
        For example:
      </p>
<pre class="programlisting"><span class="comment">// define a standard normal distribution:</span>
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span> <span class="identifier">norm</span><span class="special">;</span>
<span class="comment">// print the value of x for which the complement</span>
<span class="comment">// of the probability is 0.05:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
</pre>
<p>
        The function computes a value <span class="emphasis"><em>x</em></span> such that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
        <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code> where
        <span class="emphasis"><em>q</em></span> is complement of the probability.
      </p>
<p>
        <a class="link" href="../stat_tut/overview/complements.html#why_complements">Why complements?</a>
      </p>
<p>
        This function is also called the inverse survival function, and is the same
        as calculating the <span class="emphasis"><em>upper critical value</em></span> of a distribution.
      </p>
<p>
        This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the probablity lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
        if there is no finite value that has the specified probability.
      </p>
<p>
        The following graph show the inverse survival function for the normal distribution:
      </p>
<p>
        <span class="inlinemediaobject"><img src="../../../graphs/survival_inv.png"></span>
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h13"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.sd"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">Standard
        Deviation</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the standard deviation of distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined standard deviation.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h14"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.support"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">support</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the supported range of random variable over the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        The distribution is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the
        smallest closed set whose complement has probability zero"</a>.
        Non-mathematicians might say it means the 'interesting' smallest range of
        random variate x that has the cdf going from zero to unity. Outside are uninteresting
        zones where the pdf is zero, and the cdf zero or unity.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h15"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.variance"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">Variance</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the variance of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined variance.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h16"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.skewness"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">Skewness</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">skewness</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the skewness of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined skewness.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h17"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">Kurtosis</a>
      </h5>
<pre class="programlisting"><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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">kurtosis</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the 'proper' kurtosis (normalized fourth moment) of the distribution
        <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        kertosis = &#946;<sub>2</sub> &#160;= &#956;<sub>4</sub> &#160; / &#956;<sub>2</sub><sup>2</sup>
      </p>
<p>
        Where &#956;<sub>i</sub> &#160; is the i'th central moment of the distribution, and in particular
        &#956;<sub>2</sub> &#160; is the variance of the distribution.
      </p>
<p>
        The kurtosis is a measure of the "peakedness" of a distribution.
      </p>
<p>
        Note that the literature definition of kurtosis is confusing. The definition
        used here is that used by for example <a href="http://mathworld.wolfram.com/Kurtosis.html" target="_top">Wolfram
        MathWorld</a> (that includes a table of formulae for kurtosis excess
        for various distributions) but NOT the definition of <a href="http://en.wikipedia.org/wiki/Kurtosis" target="_top">kurtosis
        used by Wikipedia</a> which treats "kurtosis" and "kurtosis
        excess" as the same quantity.
      </p>
<pre class="programlisting"><span class="identifier">kurtosis_excess</span> <span class="special">=</span> <span class="char">'proper'</span> <span class="identifier">kurtosis</span> <span class="special">-</span> <span class="number">3</span>
</pre>
<p>
        This subtraction of 3 is convenient so that the <span class="emphasis"><em>kurtosis excess</em></span>
        of a normal distribution is zero.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined kurtosis.
      </p>
<p>
        'Proper' kurtosis can have a value from zero to + infinity.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h18"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis_excess"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">Kurtosis
        excess</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">RealType</span> <span class="identifier">kurtosis_excess</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
</pre>
<p>
        Returns the kurtosis excess of the distribution <span class="emphasis"><em>dist</em></span>.
      </p>
<p>
        kurtosis excess = &#947;<sub>2</sub> &#160;= &#956;<sub>4</sub> &#160; / &#956;<sub>2</sub><sup>2</sup> &#160;- 3 = kurtosis - 3
      </p>
<p>
        Where &#956;<sub>i</sub> &#160; is the i'th central moment of the distribution, and in particular
        &#956;<sub>2</sub> &#160; is the variance of the distribution.
      </p>
<p>
        The kurtosis excess is a measure of the "peakedness" of a distribution,
        and is more widely used than the "kurtosis proper". It is defined
        so that the kurtosis excess of a normal distribution is zero.
      </p>
<p>
        This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
        if the distribution does not have a defined kurtosis excess.
      </p>
<p>
        Kurtosis excess can have a value from -2 to + infinity.
      </p>
<pre class="programlisting"><span class="identifier">kurtosis</span> <span class="special">=</span> <span class="identifier">kurtosis_excess</span> <span class="special">+</span><span class="number">3</span><span class="special">;</span>
</pre>
<p>
        The kurtosis excess of a normal distribution is zero.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h19"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdfPQ"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P
        and Q</a>
      </h5>
<p>
        The terms P and Q are sometimes used to refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
        Distribution Function</a> and its <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">complement</a>
        respectively. Lowercase p and q are sometimes used to refer to the values
        returned by these functions.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h20"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.percent"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent
        Point Function or Percentile</a>
      </h5>
<p>
        The percent point function, also known as the percentile, is the same as
        the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h21"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse
        CDF Function.</a>
      </h5>
<p>
        The inverse of the cumulative distribution function, is the same as the
        <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h22"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse
        Survival Function.</a>
      </h5>
<p>
        The inverse of the survival function, is the same as computing the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the complement
        of the probability</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h23"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pmf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability
        Mass Function</a>
      </h5>
<p>
        The Probability Mass Function is the same as the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
        Density Function</a>.
      </p>
<p>
        The term Mass Function is usually applied to discrete distributions, while
        the term <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
        Function</a> applies to continuous distributions.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h24"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.lower_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower
        Critical Value.</a>
      </h5>
<p>
        The lower critical value calculates the value of the random variable given
        the area under the left tail of the distribution. It is equivalent to calculating
        the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h25"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.upper_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper
        Critical Value.</a>
      </h5>
<p>
        The upper critical value calculates the value of the random variable given
        the area under the right tail of the distribution. It is equivalent to calculating
        the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the
        complement of the probability</a>.
      </p>
<h5>
<a name="math_toolkit.dist_ref.nmp.h26"></a>
        <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival
        Function</a>
      </h5>
<p>
        Refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of
        the Cumulative Distribution Function</a>.
      </p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
      Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="../dist_ref.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.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="dists.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>