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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)
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>
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>
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>
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>
55 <span class="special">}}</span> <span class="comment">// namespaces</span>
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>.
64 Given mean μ  and standard deviation σ  it has the PDF:
67   <span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.svg"></span>
70 The variation the PDF with its parameters is illustrated in the following
74 <span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.svg" align="middle"></span>
77 The cumulative distribution function is given by
80   <span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.svg"></span>
83 and illustrated by this graph
86 <span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.svg" align="middle"></span>
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
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>
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>.
100 Requires sd > 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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>
107 both return the <span class="emphasis"><em>mean</em></span> of this distribution.
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>
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
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
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>.
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.
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>
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.
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>
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.
156 <div class="informaltable"><table class="table">
182 Using the relation: pdf = e<sup>-(x-m)<sup>2</sup>/(2s<sup>2</sup>)</sup> / (s * sqrt(2*pi))
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)))
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)))
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)
225 quantile from the complement
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)
237 mean and standard deviation
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>
254 The same as the mean.
266 The same as the mean.
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>)
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