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  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">&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>
  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">&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>
  34           <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>
  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">&lt;&gt;</span> <span class="identifier">normal</span><span class="special">;</span>
  38
  39 <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>
  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 &#956; &#160;and standard deviation &#963; &#160;it has the PDF:
  65         </p>
  66 <p>
  67           &#160;   <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           &#160;   <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 &gt; 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 +&#8734; and -&#8734; = 0 is also supported, and cdf at -&#8734; = 0, cdf at +&#8734; = 1, and
 138           complement cdf -&#8734; = 1 and +&#8734; = 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 &#169; 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 R&#229;de, 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>
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