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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.exp_dist"></a><a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>
  28 </h4></div></div></div>
  29 <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>
  30 <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>
  31           <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>
  32 <span class="keyword">class</span> <span class="identifier">exponential_distribution</span><span class="special">;</span>
  33
  34 <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>
  35
  36 <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>
  37 <span class="keyword">class</span> <span class="identifier">exponential_distribution</span>
  38 <span class="special">{</span>
  39 <span class="keyword">public</span><span class="special">:</span>
  40    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  41    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  42
  43    <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>
  44
  45    <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  46 <span class="special">};</span>
  47 </pre>
  48 <p>
  49           The <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential
  50           distribution</a> is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous
  51           probability distribution</a> with PDF:
  52         </p>
  53 <p>
  54           <span class="inlinemediaobject"><img src="../../../../equations/exponential_dist_ref1.svg"></span>
  55         </p>
  56 <p>
  57           It is often used to model the time between independent events that happen
  58           at a constant average rate.
  59         </p>
  60 <p>
  61           The following graph shows how the distribution changes for different values
  62           of the rate parameter lambda:
  63         </p>
  64 <p>
  65           <span class="inlinemediaobject"><img src="../../../../graphs/exponential_pdf.svg" align="middle"></span>
  66         </p>
  67 <h5>
  68 <a name="math_toolkit.dist_ref.dists.exp_dist.h0"></a>
  69           <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.member_functions"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.member_functions">Member
  70           Functions</a>
  71         </h5>
  72 <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>
  73 </pre>
  74 <p>
  75           Constructs an <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Exponential
  76           distribution</a> with parameter <span class="emphasis"><em>lambda</em></span>. Lambda
  77           is defined as the reciprocal of the scale parameter.
  78         </p>
  79 <p>
  80           Requires lambda &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
  81         </p>
  82 <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>
  83 </pre>
  84 <p>
  85           Accessor function returns the lambda parameter of the distribution.
  86         </p>
  87 <h5>
  88 <a name="math_toolkit.dist_ref.dists.exp_dist.h1"></a>
  89           <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.non_member_accessors"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.non_member_accessors">Non-member
  90           Accessors</a>
  91         </h5>
  92 <p>
  93           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
  94           functions</a> that are generic to all distributions are supported:
  95           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
  96           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
  97           <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>,
  98           <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>,
  99           <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>,
 100           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 101           <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>,
 102           <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>.
 103         </p>
 104 <p>
 105           The domain of the random variable is [0, +&#8734;].
 106         </p>
 107 <h5>
 108 <a name="math_toolkit.dist_ref.dists.exp_dist.h2"></a>
 109           <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.accuracy"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.accuracy">Accuracy</a>
 110         </h5>
 111 <p>
 112           The exponential distribution is implemented in terms of the standard library
 113           functions <code class="computeroutput"><span class="identifier">exp</span></code>, <code class="computeroutput"><span class="identifier">log</span></code>, <code class="computeroutput"><span class="identifier">log1p</span></code>
 114           and <code class="computeroutput"><span class="identifier">expm1</span></code> and as such should
 115           have very low error rates.
 116         </p>
 117 <h5>
 118 <a name="math_toolkit.dist_ref.dists.exp_dist.h3"></a>
 119           <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.implementation"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.implementation">Implementation</a>
 120         </h5>
 121 <p>
 122           In the following table &#955; is the parameter lambda of the distribution, <span class="emphasis"><em>x</em></span>
 123           is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 124           = 1-p</em></span>.
 125         </p>
 126 <div class="informaltable"><table class="table">
 127 <colgroup>
 128 <col>
 129 <col>
 130 </colgroup>
 131 <thead><tr>
 132 <th>
 133                   <p>
 134                     Function
 135                   </p>
 136                 </th>
 137 <th>
 138                   <p>
 139                     Implementation Notes
 140                   </p>
 141                 </th>
 142 </tr></thead>
 143 <tbody>
 144 <tr>
 145 <td>
 146                   <p>
 147                     pdf
 148                   </p>
 149                 </td>
 150 <td>
 151                   <p>
 152                     Using the relation: pdf = &#955; * exp(-&#955; * x)
 153                   </p>
 154                 </td>
 155 </tr>
 156 <tr>
 157 <td>
 158                   <p>
 159                     cdf
 160                   </p>
 161                 </td>
 162 <td>
 163                   <p>
 164                     Using the relation: p = 1 - exp(-x * &#955;) = -expm1(-x * &#955;)
 165                   </p>
 166                 </td>
 167 </tr>
 168 <tr>
 169 <td>
 170                   <p>
 171                     cdf complement
 172                   </p>
 173                 </td>
 174 <td>
 175                   <p>
 176                     Using the relation: q = exp(-x * &#955;)
 177                   </p>
 178                 </td>
 179 </tr>
 180 <tr>
 181 <td>
 182                   <p>
 183                     quantile
 184                   </p>
 185                 </td>
 186 <td>
 187                   <p>
 188                     Using the relation: x = -log(1-p) / &#955; = -log1p(-p) / &#955;
 189                   </p>
 190                 </td>
 191 </tr>
 192 <tr>
 193 <td>
 194                   <p>
 195                     quantile from the complement
 196                   </p>
 197                 </td>
 198 <td>
 199                   <p>
 200                     Using the relation: x = -log(q) / &#955;
 201                   </p>
 202                 </td>
 203 </tr>
 204 <tr>
 205 <td>
 206                   <p>
 207                     mean
 208                   </p>
 209                 </td>
 210 <td>
 211                   <p>
 212                     1/&#955;
 213                   </p>
 214                 </td>
 215 </tr>
 216 <tr>
 217 <td>
 218                   <p>
 219                     standard deviation
 220                   </p>
 221                 </td>
 222 <td>
 223                   <p>
 224                     1/&#955;
 225                   </p>
 226                 </td>
 227 </tr>
 228 <tr>
 229 <td>
 230                   <p>
 231                     mode
 232                   </p>
 233                 </td>
 234 <td>
 235                   <p>
 236                     0
 237                   </p>
 238                 </td>
 239 </tr>
 240 <tr>
 241 <td>
 242                   <p>
 243                     skewness
 244                   </p>
 245                 </td>
 246 <td>
 247                   <p>
 248                     2
 249                   </p>
 250                 </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                   <p>
 255                     kurtosis
 256                   </p>
 257                 </td>
 258 <td>
 259                   <p>
 260                     9
 261                   </p>
 262                 </td>
 263 </tr>
 264 <tr>
 265 <td>
 266                   <p>
 267                     kurtosis excess
 268                   </p>
 269                 </td>
 270 <td>
 271                   <p>
 272                     6
 273                   </p>
 274                 </td>
 275 </tr>
 276 </tbody>
 277 </table></div>
 278 <h5>
 279 <a name="math_toolkit.dist_ref.dists.exp_dist.h4"></a>
 280           <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.references"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.references">references</a>
 281         </h5>
 282 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 283 <li class="listitem">
 284               <a href="http://mathworld.wolfram.com/ExponentialDistribution.html" target="_top">Weisstein,
 285               Eric W. "Exponential Distribution." From MathWorld--A Wolfram
 286               Web Resource</a>
 287             </li>
 288 <li class="listitem">
 289               <a href="http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html" target="_top">Wolfram
 290               Mathematica calculator</a>
 291             </li>
 292 <li class="listitem">
 293               <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm" target="_top">NIST
 294               Exploratory Data Analysis</a>
 295             </li>
 296 <li class="listitem">
 297               <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Wikipedia
 298               Exponential distribution</a>
 299             </li>
 300 </ul></div>
 301 <p>
 302           (See also the reference documentation for the related <a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme
 303           Distributions</a>.)
 304         </p>
 305 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 306               <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme
 307               Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
 308               Nadarajah</a> discuss the relationship of the types of extreme
 309               value distributions.
 310             </li></ul></div>
 311 </div>
 312 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 313 <td align="left"></td>
 314 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 315       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 316       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 317       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 318         Distributed under the Boost Software License, Version 1.0. (See accompanying
 319         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>)
 320       </p>
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