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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;20.&#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;20.&#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; and standard deviation &#963; it has the PDF:
  65         </p>
  66 <div class="blockquote"><blockquote class="blockquote"><p>
  67             <span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.svg"></span>
  68
  69           </p></blockquote></div>
  70 <p>
  71           The variation the PDF with its parameters is illustrated in the following
  72           graph:
  73         </p>
  74 <div class="blockquote"><blockquote class="blockquote"><p>
  75             <span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.svg" align="middle"></span>
  76
  77           </p></blockquote></div>
  78 <p>
  79           The cumulative distribution function is given by
  80         </p>
  81 <div class="blockquote"><blockquote class="blockquote"><p>
  82             <span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.svg"></span>
  83
  84           </p></blockquote></div>
  85 <p>
  86           and illustrated by this graph
  87         </p>
  88 <div class="blockquote"><blockquote class="blockquote"><p>
  89             <span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.svg" align="middle"></span>
  90
  91           </p></blockquote></div>
  92 <h5>
  93 <a name="math_toolkit.dist_ref.dists.normal_dist.h0"></a>
  94           <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
  95           Functions</a>
  96         </h5>
  97 <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>
  98 </pre>
  99 <p>
 100           Constructs a normal distribution with mean <span class="emphasis"><em>mean</em></span> and
 101           standard deviation <span class="emphasis"><em>sd</em></span>.
 102         </p>
 103 <p>
 104           Requires <span class="emphasis"><em>sd</em></span> &gt; 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 105           is called.
 106         </p>
 107 <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>
 108 <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 109 </pre>
 110 <p>
 111           both return the <span class="emphasis"><em>mean</em></span> of this distribution.
 112         </p>
 113 <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>
 114 <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 115 </pre>
 116 <p>
 117           both return the <span class="emphasis"><em>standard deviation</em></span> of this distribution.
 118           (Redundant location and scale function are provided to match other similar
 119           distributions, allowing the functions find_location and find_scale to be
 120           used generically).
 121         </p>
 122 <h5>
 123 <a name="math_toolkit.dist_ref.dists.normal_dist.h1"></a>
 124           <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
 125           Accessors</a>
 126         </h5>
 127 <p>
 128           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 129           functions</a> that are generic to all distributions are supported:
 130           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 131           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 132           <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>,
 133           <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>,
 134           <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>,
 135           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 136           <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>,
 137           <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>.
 138         </p>
 139 <p>
 140           The domain of the random variable is [-[max_value], +[min_value]]. However,
 141           the pdf of +&#8734; and -&#8734; = 0 is also supported, and cdf at -&#8734; = 0, cdf at +&#8734; = 1, and
 142           complement cdf -&#8734; = 1 and +&#8734; = 0, if RealType permits.
 143         </p>
 144 <h5>
 145 <a name="math_toolkit.dist_ref.dists.normal_dist.h2"></a>
 146           <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>
 147         </h5>
 148 <p>
 149           The normal distribution is implemented in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">error
 150           function</a>, and as such should have very low error rates.
 151         </p>
 152 <h5>
 153 <a name="math_toolkit.dist_ref.dists.normal_dist.h3"></a>
 154           <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>
 155         </h5>
 156 <p>
 157           In the following table <span class="emphasis"><em>m</em></span> is the mean of the distribution,
 158           and <span class="emphasis"><em>s</em></span> is its standard deviation.
 159         </p>
 160 <div class="informaltable"><table class="table">
 161 <colgroup>
 162 <col>
 163 <col>
 164 </colgroup>
 165 <thead><tr>
 166 <th>
 167                   <p>
 168                     Function
 169                   </p>
 170                 </th>
 171 <th>
 172                   <p>
 173                     Implementation Notes
 174                   </p>
 175                 </th>
 176 </tr></thead>
 177 <tbody>
 178 <tr>
 179 <td>
 180                   <p>
 181                     pdf
 182                   </p>
 183                 </td>
 184 <td>
 185                   <p>
 186                     Using the relation: pdf = e<sup>-(x-m)<sup>2</sup>/(2s<sup>2</sup>)</sup> / (s * sqrt(2*pi))
 187                   </p>
 188                 </td>
 189 </tr>
 190 <tr>
 191 <td>
 192                   <p>
 193                     cdf
 194                   </p>
 195                 </td>
 196 <td>
 197                   <p>
 198                     Using the relation: p = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">erfc</a>(-(x-m)/(s*sqrt(2)))
 199                   </p>
 200                 </td>
 201 </tr>
 202 <tr>
 203 <td>
 204                   <p>
 205                     cdf complement
 206                   </p>
 207                 </td>
 208 <td>
 209                   <p>
 210                     Using the relation: q = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">erfc</a>((x-m)/(s*sqrt(2)))
 211                   </p>
 212                 </td>
 213 </tr>
 214 <tr>
 215 <td>
 216                   <p>
 217                     quantile
 218                   </p>
 219                 </td>
 220 <td>
 221                   <p>
 222                     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)
 223                   </p>
 224                 </td>
 225 </tr>
 226 <tr>
 227 <td>
 228                   <p>
 229                     quantile from the complement
 230                   </p>
 231                 </td>
 232 <td>
 233                   <p>
 234                     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)
 235                   </p>
 236                 </td>
 237 </tr>
 238 <tr>
 239 <td>
 240                   <p>
 241                     mean and standard deviation
 242                   </p>
 243                 </td>
 244 <td>
 245                   <p>
 246                     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>
 247                   </p>
 248                 </td>
 249 </tr>
 250 <tr>
 251 <td>
 252                   <p>
 253                     mode
 254                   </p>
 255                 </td>
 256 <td>
 257                   <p>
 258                     The same as the mean.
 259                   </p>
 260                 </td>
 261 </tr>
 262 <tr>
 263 <td>
 264                   <p>
 265                     median
 266                   </p>
 267                 </td>
 268 <td>
 269                   <p>
 270                     The same as the mean.
 271                   </p>
 272                 </td>
 273 </tr>
 274 <tr>
 275 <td>
 276                   <p>
 277                     skewness
 278                   </p>
 279                 </td>
 280 <td>
 281                   <p>
 282                     0
 283                   </p>
 284                 </td>
 285 </tr>
 286 <tr>
 287 <td>
 288                   <p>
 289                     kurtosis
 290                   </p>
 291                 </td>
 292 <td>
 293                   <p>
 294                     3
 295                   </p>
 296                 </td>
 297 </tr>
 298 <tr>
 299 <td>
 300                   <p>
 301                     kurtosis excess
 302                   </p>
 303                 </td>
 304 <td>
 305                   <p>
 306                     0
 307                   </p>
 308                 </td>
 309 </tr>
 310 </tbody>
 311 </table></div>
 312 </div>
 313 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 314 <td align="left"></td>
 315 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 316       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 317       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 318       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 319       Daryle Walker and Xiaogang Zhang<p>
 320         Distributed under the Boost Software License, Version 1.0. (See accompanying
 321         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>)
 322       </p>
 323 </div></td>
 324 </tr></table>
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