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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.normal_dist"></a><a class="link" href="normal_dist.html" title="Normal (Gaussian) Distribution">Normal (Gaussian)
        Distribution</a>
</h4></div></div></div>
<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">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<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="Chapter&#160;15.&#160;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&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">normal_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">normal_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">normal</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="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">normal_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">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</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">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// location.</span>
   <span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// scale.</span>
   <span class="comment">// Synonyms, provided to allow generic use of find_location and find_scale.</span>
   <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
   <span class="identifier">RealType</span> <span class="identifier">scale</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 normal distribution is probably the most well known statistical distribution:
          it is also known as the Gaussian Distribution. A normal distribution with
          mean zero and standard deviation one is known as the <span class="emphasis"><em>Standard
          Normal Distribution</em></span>.
        </p>
<p>
          Given mean &#956; &#160;and standard deviation &#963; &#160;it has the PDF:
        </p>
<p>
          &#160;   <span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.svg"></span>
        </p>
<p>
          The variation the PDF with its parameters is illustrated in the following
          graph:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.svg" align="middle"></span>
        </p>
<p>
          The cumulative distribution function is given by
        </p>
<p>
          &#160;   <span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.svg"></span>
        </p>
<p>
          and illustrated by this graph
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.svg" align="middle"></span>
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.normal_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.member_functions"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs a normal distribution with mean <span class="emphasis"><em>mean</em></span> and
          standard deviation <span class="emphasis"><em>sd</em></span>.
        </p>
<p>
          Requires sd &gt; 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
          is called.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          both return the <span class="emphasis"><em>mean</em></span> of this distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          both return the <span class="emphasis"><em>standard deviation</em></span> of this distribution.
          (Redundant location and scale function are provided to match other similar
          distributions, allowing the functions find_location and find_scale to be
          used generically).
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.normal_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.non_member_accessors"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <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>,
          <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>,
          <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>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <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>.
        </p>
<p>
          The domain of the random variable is [-[max_value], +[min_value]]. However,
          the pdf of +&#8734; and -&#8734; = 0 is also supported, and cdf at -&#8734; = 0, cdf at +&#8734; = 1, and
          complement cdf -&#8734; = 1 and +&#8734; = 0, if RealType permits.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.normal_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.accuracy"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The normal distribution is implemented in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">error
          function</a>, and as such should have very low error rates.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.normal_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.implementation"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table <span class="emphasis"><em>m</em></span> is the mean of the distribution,
          and <span class="emphasis"><em>s</em></span> is its standard deviation.
        </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 = e<sup>-(x-m)<sup>2</sup>/(2s<sup>2</sup>)</sup> / (s * sqrt(2*pi))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">erfc</a>(-(x-m)/(s*sqrt(2)))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">erfc</a>((x-m)/(s*sqrt(2)))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = m - s * sqrt(2) * <a class="link" href="../../sf_erf/error_inv.html" title="Error Function Inverses">erfc_inv</a>(2*p)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = m + s * sqrt(2) * <a class="link" href="../../sf_erf/error_inv.html" title="Error Function Inverses">erfc_inv</a>(2*p)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean and standard deviation
                  </p>
                </td>
<td>
                  <p>
                    The same as <code class="computeroutput"><span class="identifier">dist</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span></code> and <code class="computeroutput"><span class="identifier">dist</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span></code>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    The same as the mean.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    median
                  </p>
                </td>
<td>
                  <p>
                    The same as the mean.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    3
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    0
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</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>
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	24	</div>
	25	<div class="section">
	26	<div class="titlepage"><div><div><h4 class="title">
	27	<a name="math_toolkit.dist_ref.dists.normal_dist"></a><a class="link" href="normal_dist.html" title="Normal (Gaussian) Distribution">Normal (Gaussian)
	28	Distribution</a>
	29	</h4></div></div></div>
	30	<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">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
	31	<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>
	32
	33	<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>
	34	<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
	35	<span class="keyword">class</span> <span class="identifier">normal_distribution</span><span class="special">;</span>
	36
	37	<span class="keyword">typedef</span> <span class="identifier">normal_distribution</span><span class="special"><></span> <span class="identifier">normal</span><span class="special">;</span>
	38
	39	<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
	40	<span class="keyword">class</span> <span class="identifier">normal_distribution</span>
	41	<span class="special">{</span>
	42	<span class="keyword">public</span><span class="special">:</span>
	43	<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
	44	<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
	45	<span class="comment">// Construct:</span>
	46	<span class="identifier">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
	47	<span class="comment">// Accessors:</span>
	48	<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// location.</span>
	49	<span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// scale.</span>
	50	<span class="comment">// Synonyms, provided to allow generic use of find_location and find_scale.</span>
	51	<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
	52	<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
	53	<span class="special">};</span>
	54
	55	<span class="special">}}</span> <span class="comment">// namespaces</span>
	56	</pre>
	57	<p>
	58	The normal distribution is probably the most well known statistical distribution:
	59	it is also known as the Gaussian Distribution. A normal distribution with
	60	mean zero and standard deviation one is known as the <span class="emphasis"><em>Standard
	61	Normal Distribution</em></span>.
	62	</p>
	63	<p>
	64	Given mean μ  and standard deviation σ  it has the PDF:
65	</p>
66	<p>
67	<span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.svg"></span>
68	</p>
69	<p>
70	The variation the PDF with its parameters is illustrated in the following
71	graph:
72	</p>
73	<p>
74	<span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.svg" align="middle"></span>
75	</p>
76	<p>
77	The cumulative distribution function is given by
78	</p>
79	<p>
80	<span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.svg"></span>
81	</p>
82	<p>
83	and illustrated by this graph
84	</p>
85	<p>
86	<span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.svg" align="middle"></span>
87	</p>
88	<h5>
89	<a name="math_toolkit.dist_ref.dists.normal_dist.h0"></a>
90	<span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.member_functions"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.member_functions">Member
91	Functions</a>
92	</h5>
93	<pre class="programlisting"><span class="identifier">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
94	</pre>
95	<p>
96	Constructs a normal distribution with mean <span class="emphasis"><em>mean</em></span> and
97	standard deviation <span class="emphasis"><em>sd</em></span>.
98	</p>
99	<p>
100	Requires sd > 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
101	is called.
102	</p>
103	<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
104	<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
105	</pre>
106	<p>
107	both return the <span class="emphasis"><em>mean</em></span> of this distribution.
108	</p>
109	<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
110	<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
111	</pre>
112	<p>
113	both return the <span class="emphasis"><em>standard deviation</em></span> of this distribution.
114	(Redundant location and scale function are provided to match other similar
115	distributions, allowing the functions find_location and find_scale to be
116	used generically).
117	</p>
118	<h5>
119	<a name="math_toolkit.dist_ref.dists.normal_dist.h1"></a>
120	<span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.non_member_accessors"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.non_member_accessors">Non-member
121	Accessors</a>
122	</h5>
123	<p>
124	All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
125	functions</a> that are generic to all distributions are supported:
126	<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
127	<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
128	<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>,
129	<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>,
130	<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>,
131	<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
132	<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>,
133	<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>.
134	</p>
135	<p>
136	The domain of the random variable is [-[max_value], +[min_value]]. However,
137	the pdf of +∞ and -∞ = 0 is also supported, and cdf at -∞ = 0, cdf at +∞ = 1, and
138	complement cdf -∞ = 1 and +∞ = 0, if RealType permits.
139	</p>
140	<h5>
141	<a name="math_toolkit.dist_ref.dists.normal_dist.h2"></a>
142	<span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.accuracy"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.accuracy">Accuracy</a>
143	</h5>
144	<p>
145	The normal distribution is implemented in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">error
146	function</a>, and as such should have very low error rates.
147	</p>
148	<h5>
149	<a name="math_toolkit.dist_ref.dists.normal_dist.h3"></a>
150	<span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.implementation"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.implementation">Implementation</a>
151	</h5>
152	<p>
153	In the following table <span class="emphasis"><em>m</em></span> is the mean of the distribution,
154	and <span class="emphasis"><em>s</em></span> is its standard deviation.
155	</p>
156	<div class="informaltable"><table class="table">
157	<colgroup>
158	<col>
159	<col>
160	</colgroup>
161	<thead><tr>
162	<th>
163	<p>
164	Function
165	</p>
166	</th>
167	<th>
168	<p>
169	Implementation Notes
170	</p>
171	</th>
172	</tr></thead>
173	<tbody>
174	<tr>
175	<td>
176	<p>
177	pdf
178	</p>
179	</td>
180	<td>
181	<p>
182	Using the relation: pdf = e<sup>-(x-m)<sup>2</sup>/(2s<sup>2</sup>)</sup> / (s * sqrt(2*pi))
183	</p>
184	</td>
185	</tr>
186	<tr>
187	<td>
188	<p>
189	cdf
190	</p>
191	</td>
192	<td>
193	<p>
194	Using the relation: p = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">erfc</a>(-(x-m)/(s*sqrt(2)))
195	</p>
196	</td>
197	</tr>
198	<tr>
199	<td>
200	<p>
201	cdf complement
202	</p>
203	</td>
204	<td>
205	<p>
206	Using the relation: q = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">erfc</a>((x-m)/(s*sqrt(2)))
207	</p>
208	</td>
209	</tr>
210	<tr>
211	<td>
212	<p>
213	quantile
214	</p>
215	</td>
216	<td>
217	<p>
218	Using the relation: x = m - s * sqrt(2) * <a class="link" href="../../sf_erf/error_inv.html" title="Error Function Inverses">erfc_inv</a>(2*p)
219	</p>
220	</td>
221	</tr>
222	<tr>
223	<td>
224	<p>
225	quantile from the complement
226	</p>
227	</td>
228	<td>
229	<p>
230	Using the relation: x = m + s * sqrt(2) * <a class="link" href="../../sf_erf/error_inv.html" title="Error Function Inverses">erfc_inv</a>(2*p)
231	</p>
232	</td>
233	</tr>
234	<tr>
235	<td>
236	<p>
237	mean and standard deviation
238	</p>
239	</td>
240	<td>
241	<p>
242	The same as <code class="computeroutput"><span class="identifier">dist</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span></code> and <code class="computeroutput"><span class="identifier">dist</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span></code>
243	</p>
244	</td>
245	</tr>
246	<tr>
247	<td>
248	<p>
249	mode
250	</p>
251	</td>
252	<td>
253	<p>
254	The same as the mean.
255	</p>
256	</td>
257	</tr>
258	<tr>
259	<td>
260	<p>
261	median
262	</p>
263	</td>
264	<td>
265	<p>
266	The same as the mean.
267	</p>
268	</td>
269	</tr>
270	<tr>
271	<td>
272	<p>
273	skewness
274	</p>
275	</td>
276	<td>
277	<p>
278	0
279	</p>
280	</td>
281	</tr>
282	<tr>
283	<td>
284	<p>
285	kurtosis
286	</p>
287	</td>
288	<td>
289	<p>
290	3
291	</p>
292	</td>
293	</tr>
294	<tr>
295	<td>
296	<p>
297	kurtosis excess
298	</p>
299	</td>
300	<td>
301	<p>
302	0
303	</p>
304	</td>
305	</tr>
306	</tbody>
307	</table></div>
308	</div>
309	<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
310	<td align="left"></td>
311	<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
312	Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
313	Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
314	Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
315	Distributed under the Boost Software License, Version 1.0. (See accompanying
316	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>)
317	</p>
318	</div></td>
319	</tr></table>
320	<hr>
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