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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;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>
  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;20.&#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 <div class="blockquote"><blockquote class="blockquote"><p>
  54             <span class="inlinemediaobject"><img src="../../../../equations/exponential_dist_ref1.svg"></span>
  55
  56           </p></blockquote></div>
  57 <p>
  58           It is often used to model the time between independent events that happen
  59           at a constant average rate.
  60         </p>
  61 <p>
  62           The following graph shows how the distribution changes for different values
  63           of the rate parameter lambda:
  64         </p>
  65 <div class="blockquote"><blockquote class="blockquote"><p>
  66             <span class="inlinemediaobject"><img src="../../../../graphs/exponential_pdf.svg" align="middle"></span>
  67
  68           </p></blockquote></div>
  69 <h5>
  70 <a name="math_toolkit.dist_ref.dists.exp_dist.h0"></a>
  71           <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
  72           Functions</a>
  73         </h5>
  74 <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>
  75 </pre>
  76 <p>
  77           Constructs an <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Exponential
  78           distribution</a> with parameter <span class="emphasis"><em>lambda</em></span>. Lambda
  79           is defined as the reciprocal of the scale parameter.
  80         </p>
  81 <p>
  82           Requires lambda &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
  83         </p>
  84 <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>
  85 </pre>
  86 <p>
  87           Accessor function returns the lambda parameter of the distribution.
  88         </p>
  89 <h5>
  90 <a name="math_toolkit.dist_ref.dists.exp_dist.h1"></a>
  91           <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
  92           Accessors</a>
  93         </h5>
  94 <p>
  95           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
  96           functions</a> that are generic to all distributions are supported:
  97           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
  98           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
  99           <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>,
 100           <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>,
 101           <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>,
 102           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 103           <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>,
 104           <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>.
 105         </p>
 106 <p>
 107           The domain of the random variable is [0, +&#8734;].
 108         </p>
 109 <h5>
 110 <a name="math_toolkit.dist_ref.dists.exp_dist.h2"></a>
 111           <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>
 112         </h5>
 113 <p>
 114           The exponential distribution is implemented in terms of the standard library
 115           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>
 116           and <code class="computeroutput"><span class="identifier">expm1</span></code> and as such should
 117           have very low error rates.
 118         </p>
 119 <h5>
 120 <a name="math_toolkit.dist_ref.dists.exp_dist.h3"></a>
 121           <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>
 122         </h5>
 123 <p>
 124           In the following table &#955; is the parameter lambda of the distribution, <span class="emphasis"><em>x</em></span>
 125           is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 126           = 1-p</em></span>.
 127         </p>
 128 <div class="informaltable"><table class="table">
 129 <colgroup>
 130 <col>
 131 <col>
 132 </colgroup>
 133 <thead><tr>
 134 <th>
 135                   <p>
 136                     Function
 137                   </p>
 138                 </th>
 139 <th>
 140                   <p>
 141                     Implementation Notes
 142                   </p>
 143                 </th>
 144 </tr></thead>
 145 <tbody>
 146 <tr>
 147 <td>
 148                   <p>
 149                     pdf
 150                   </p>
 151                 </td>
 152 <td>
 153                   <p>
 154                     Using the relation: pdf = &#955; * exp(-&#955; * x)
 155                   </p>
 156                 </td>
 157 </tr>
 158 <tr>
 159 <td>
 160                   <p>
 161                     cdf
 162                   </p>
 163                 </td>
 164 <td>
 165                   <p>
 166                     Using the relation: p = 1 - exp(-x * &#955;) = -expm1(-x * &#955;)
 167                   </p>
 168                 </td>
 169 </tr>
 170 <tr>
 171 <td>
 172                   <p>
 173                     cdf complement
 174                   </p>
 175                 </td>
 176 <td>
 177                   <p>
 178                     Using the relation: q = exp(-x * &#955;)
 179                   </p>
 180                 </td>
 181 </tr>
 182 <tr>
 183 <td>
 184                   <p>
 185                     quantile
 186                   </p>
 187                 </td>
 188 <td>
 189                   <p>
 190                     Using the relation: x = -log(1-p) / &#955; = -log1p(-p) / &#955;
 191                   </p>
 192                 </td>
 193 </tr>
 194 <tr>
 195 <td>
 196                   <p>
 197                     quantile from the complement
 198                   </p>
 199                 </td>
 200 <td>
 201                   <p>
 202                     Using the relation: x = -log(q) / &#955;
 203                   </p>
 204                 </td>
 205 </tr>
 206 <tr>
 207 <td>
 208                   <p>
 209                     mean
 210                   </p>
 211                 </td>
 212 <td>
 213                   <p>
 214                     1/&#955;
 215                   </p>
 216                 </td>
 217 </tr>
 218 <tr>
 219 <td>
 220                   <p>
 221                     standard deviation
 222                   </p>
 223                 </td>
 224 <td>
 225                   <p>
 226                     1/&#955;
 227                   </p>
 228                 </td>
 229 </tr>
 230 <tr>
 231 <td>
 232                   <p>
 233                     mode
 234                   </p>
 235                 </td>
 236 <td>
 237                   <p>
 238                     0
 239                   </p>
 240                 </td>
 241 </tr>
 242 <tr>
 243 <td>
 244                   <p>
 245                     skewness
 246                   </p>
 247                 </td>
 248 <td>
 249                   <p>
 250                     2
 251                   </p>
 252                 </td>
 253 </tr>
 254 <tr>
 255 <td>
 256                   <p>
 257                     kurtosis
 258                   </p>
 259                 </td>
 260 <td>
 261                   <p>
 262                     9
 263                   </p>
 264                 </td>
 265 </tr>
 266 <tr>
 267 <td>
 268                   <p>
 269                     kurtosis excess
 270                   </p>
 271                 </td>
 272 <td>
 273                   <p>
 274                     6
 275                   </p>
 276                 </td>
 277 </tr>
 278 </tbody>
 279 </table></div>
 280 <h5>
 281 <a name="math_toolkit.dist_ref.dists.exp_dist.h4"></a>
 282           <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>
 283         </h5>
 284 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 285 <li class="listitem">
 286               <a href="http://mathworld.wolfram.com/ExponentialDistribution.html" target="_top">Weisstein,
 287               Eric W. "Exponential Distribution." From MathWorld--A Wolfram
 288               Web Resource</a>
 289             </li>
 290 <li class="listitem">
 291               <a href="http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html" target="_top">Wolfram
 292               Mathematica calculator</a>
 293             </li>
 294 <li class="listitem">
 295               <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm" target="_top">NIST
 296               Exploratory Data Analysis</a>
 297             </li>
 298 <li class="listitem">
 299               <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Wikipedia
 300               Exponential distribution</a>
 301             </li>
 302 </ul></div>
 303 <p>
 304           (See also the reference documentation for the related <a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme
 305           Distributions</a>.)
 306         </p>
 307 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 308               <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme
 309               Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
 310               Nadarajah</a> discuss the relationship of the types of extreme
 311               value distributions.
 312             </li></ul></div>
 313 </div>
 314 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 315 <td align="left"></td>
 316 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 317       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 318       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 319       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 320       Daryle Walker and Xiaogang Zhang<p>
 321         Distributed under the Boost Software License, Version 1.0. (See accompanying
 322         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>)
 323       </p>
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