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  25 <div class="section math_toolkit_dist_dist_ref_dists_exp_dist">
  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.dist.dist_ref.dists.exp_dist"></a><a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential
  28           Distribution</a>
  29 </h5></div></div></div>
  30 <p>
  31 </p>
  32 <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">exponential</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  33 <p>
  34           </p>
  35 <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">double</span><span class="special">,</span>
  36           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a>   <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  37 <span class="keyword">class</span> <span class="identifier">exponential_distribution</span><span class="special">;</span>
  38
  39 <span class="keyword">typedef</span> <span class="identifier">exponential_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">exponential</span><span class="special">;</span>
  40
  41 <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="Policies">Policy</a><span class="special">&gt;</span>
  42 <span class="keyword">class</span> <span class="identifier">exponential_distribution</span>
  43 <span class="special">{</span>
  44 <span class="keyword">public</span><span class="special">:</span>
  45    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  46    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  47
  48    <span class="identifier">exponential_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
  49
  50    <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  51 <span class="special">};</span>
  52 </pre>
  53 <p>
  54             The <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential
  55             distribution</a> is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous
  56             probability distribution</a> with PDF:
  57           </p>
  58 <p>
  59             <span class="inlinemediaobject"><img src="../../../../../equations/exponential_dist_ref1.png"></span>
  60           </p>
  61 <p>
  62             It is often used to model the time between independent events that happen
  63             at a constant average rate.
  64           </p>
  65 <p>
  66             The following graph shows how the distribution changes for different
  67             values of the rate parameter lambda:
  68           </p>
  69 <p>
  70             <span class="inlinemediaobject"><img src="../../../../../graphs/exponential_pdf.png" align="middle"></span>
  71           </p>
  72 <h5>
  73 <a name="math_toolkit.dist.dist_ref.dists.exp_dist.h0"></a>
  74             <span><a name="math_toolkit.dist.dist_ref.dists.exp_dist.member_functions"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.member_functions">Member
  75             Functions</a>
  76           </h5>
  77 <pre class="programlisting"><span class="identifier">exponential_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
  78 </pre>
  79 <p>
  80             Constructs an <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Exponential
  81             distribution</a> with parameter <span class="emphasis"><em>lambda</em></span>. Lambda
  82             is defined as the reciprocal of the scale parameter.
  83           </p>
  84 <p>
  85             Requires lambda &gt; 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
  86           </p>
  87 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  88 </pre>
  89 <p>
  90             Accessor function returns the lambda parameter of the distribution.
  91           </p>
  92 <h5>
  93 <a name="math_toolkit.dist.dist_ref.dists.exp_dist.h1"></a>
  94             <span><a name="math_toolkit.dist.dist_ref.dists.exp_dist.non_member_accessors"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.non_member_accessors">Non-member
  95             Accessors</a>
  96           </h5>
  97 <p>
  98             All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
  99             accessor functions</a> that are generic to all distributions are supported:
 100             <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>,
 101             <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard
 102             Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>,
 103             <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>,
 104             <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>,
 105             <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>,
 106             <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>,
 107             <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>.
 108           </p>
 109 <p>
 110             The domain of the random variable is [0, +&#8734;].
 111           </p>
 112 <h5>
 113 <a name="math_toolkit.dist.dist_ref.dists.exp_dist.h2"></a>
 114             <span><a name="math_toolkit.dist.dist_ref.dists.exp_dist.accuracy"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.accuracy">Accuracy</a>
 115           </h5>
 116 <p>
 117             The exponential distribution is implemented in terms of the standard
 118             library functions <code class="computeroutput"><span class="identifier">exp</span></code>,
 119             <code class="computeroutput"><span class="identifier">log</span></code>, <code class="computeroutput"><span class="identifier">log1p</span></code>
 120             and <code class="computeroutput"><span class="identifier">expm1</span></code> and as such
 121             should have very low error rates.
 122           </p>
 123 <h5>
 124 <a name="math_toolkit.dist.dist_ref.dists.exp_dist.h3"></a>
 125             <span><a name="math_toolkit.dist.dist_ref.dists.exp_dist.implementation"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.implementation">Implementation</a>
 126           </h5>
 127 <p>
 128             In the following table &#955; is the parameter lambda of the distribution,
 129             <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
 130             is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 131           </p>
 132 <div class="informaltable"><table class="table">
 133 <colgroup>
 134 <col>
 135 <col>
 136 </colgroup>
 137 <thead><tr>
 138 <th>
 139                     <p>
 140                       Function
 141                     </p>
 142                   </th>
 143 <th>
 144                     <p>
 145                       Implementation Notes
 146                     </p>
 147                   </th>
 148 </tr></thead>
 149 <tbody>
 150 <tr>
 151 <td>
 152                     <p>
 153                       pdf
 154                     </p>
 155                   </td>
 156 <td>
 157                     <p>
 158                       Using the relation: pdf = &#955; * exp(-&#955; * x)
 159                     </p>
 160                   </td>
 161 </tr>
 162 <tr>
 163 <td>
 164                     <p>
 165                       cdf
 166                     </p>
 167                   </td>
 168 <td>
 169                     <p>
 170                       Using the relation: p = 1 - exp(-x * &#955;) = -expm1(-x * &#955;)
 171                     </p>
 172                   </td>
 173 </tr>
 174 <tr>
 175 <td>
 176                     <p>
 177                       cdf complement
 178                     </p>
 179                   </td>
 180 <td>
 181                     <p>
 182                       Using the relation: q = exp(-x * &#955;)
 183                     </p>
 184                   </td>
 185 </tr>
 186 <tr>
 187 <td>
 188                     <p>
 189                       quantile
 190                     </p>
 191                   </td>
 192 <td>
 193                     <p>
 194                       Using the relation: x = -log(1-p) / &#955; = -log1p(-p) / &#955;
 195                     </p>
 196                   </td>
 197 </tr>
 198 <tr>
 199 <td>
 200                     <p>
 201                       quantile from the complement
 202                     </p>
 203                   </td>
 204 <td>
 205                     <p>
 206                       Using the relation: x = -log(q) / &#955;
 207                     </p>
 208                   </td>
 209 </tr>
 210 <tr>
 211 <td>
 212                     <p>
 213                       mean
 214                     </p>
 215                   </td>
 216 <td>
 217                     <p>
 218                       1/&#955;
 219                     </p>
 220                   </td>
 221 </tr>
 222 <tr>
 223 <td>
 224                     <p>
 225                       standard deviation
 226                     </p>
 227                   </td>
 228 <td>
 229                     <p>
 230                       1/&#955;
 231                     </p>
 232                   </td>
 233 </tr>
 234 <tr>
 235 <td>
 236                     <p>
 237                       mode
 238                     </p>
 239                   </td>
 240 <td>
 241                     <p>
 242                       0
 243                     </p>
 244                   </td>
 245 </tr>
 246 <tr>
 247 <td>
 248                     <p>
 249                       skewness
 250                     </p>
 251                   </td>
 252 <td>
 253                     <p>
 254                       2
 255                     </p>
 256                   </td>
 257 </tr>
 258 <tr>
 259 <td>
 260                     <p>
 261                       kurtosis
 262                     </p>
 263                   </td>
 264 <td>
 265                     <p>
 266                       9
 267                     </p>
 268                   </td>
 269 </tr>
 270 <tr>
 271 <td>
 272                     <p>
 273                       kurtosis excess
 274                     </p>
 275                   </td>
 276 <td>
 277                     <p>
 278                       6
 279                     </p>
 280                   </td>
 281 </tr>
 282 </tbody>
 283 </table></div>
 284 <h5>
 285 <a name="math_toolkit.dist.dist_ref.dists.exp_dist.h4"></a>
 286             <span><a name="math_toolkit.dist.dist_ref.dists.exp_dist.references"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.references">references</a>
 287           </h5>
 288 <div class="itemizedlist"><ul class="itemizedlist" type="disc">
 289 <li class="listitem">
 290                 <a href="http://mathworld.wolfram.com/ExponentialDistribution.html" target="_top">Weisstein,
 291                 Eric W. "Exponential Distribution." From MathWorld--A Wolfram
 292                 Web Resource</a>
 293               </li>
 294 <li class="listitem">
 295                 <a href="http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html" target="_top">Wolfram
 296                 Mathematica calculator</a>
 297               </li>
 298 <li class="listitem">
 299                 <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm" target="_top">NIST
 300                 Exploratory Data Analysis</a>
 301               </li>
 302 <li class="listitem">
 303                 <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Wikipedia
 304                 Exponential distribution</a>
 305               </li>
 306 </ul></div>
 307 <p>
 308             (See also the reference documentation for the related <a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme
 309             Distributions</a>.)
 310           </p>
 311 <div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem">
 312                 <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme
 313                 Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
 314                 Nadarajah</a> discuss the relationship of the types of extreme
 315                 value distributions.
 316               </li></ul></div>
 317 </div>
 318 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 319 <td align="left"></td>
 320 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 321       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 322         Distributed under the Boost Software License, Version 1.0. (See accompanying
 323         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>)
 324       </p>
 325 </div></td>
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