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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.exp_dist"></a><a class="link" href="exp_dist.html" title="Exponential Distribution"> Exponential |
| Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span></pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <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<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">exponential_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">exponential_distribution</span><span class="special"><></span> <span class="identifier">exponential</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="keyword">class</span> <span class="identifier">exponential_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> |
| |
| <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> |
| |
| <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential |
| distribution</a> is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous |
| probability distribution</a> with PDF: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/exponential_dist_ref1.png"></span> |
| </p> |
| <p> |
| It is often used to model the time between independent events that happen |
| at a constant average rate. |
| </p> |
| <p> |
| The following graph shows how the distribution changes for different |
| values of the rate parameter lambda: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/exponential_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.exp_dist.member_functions"></a><h5> |
| <a name="id1021340"></a> |
| <a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <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> |
| </pre> |
| <p> |
| Constructs an <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Exponential |
| distribution</a> with parameter <span class="emphasis"><em>lambda</em></span>. Lambda |
| is defined as the reciprocal of the scale parameter. |
| </p> |
| <p> |
| Requires lambda > 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <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> |
| </pre> |
| <p> |
| Accessor function returns the lambda parameter of the distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.exp_dist.non_member_accessors"></a><h5> |
| <a name="id1021441"></a> |
| <a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, |
| <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 |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| The domain of the random variable is [0, +∞]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.exp_dist.accuracy"></a><h5> |
| <a name="id1021541"></a> |
| <a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The exponential distribution is implemented in terms of the standard |
| library 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> |
| and <code class="computeroutput"><span class="identifier">expm1</span></code> and as such |
| should have very low error rates. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.exp_dist.implementation"></a><h5> |
| <a name="id1021589"></a> |
| <a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table λ is the parameter lambda of the distribution, |
| <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> |
| is the probability and <span class="emphasis"><em>q = 1-p</em></span>. |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: pdf = λ * exp(-λ * x) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: p = 1 - exp(-x * λ) = -expm1(-x * λ) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: q = exp(-x * λ) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = -log(1-p) / λ = -log1p(-p) / λ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = -log(q) / λ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| 1/λ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| standard deviation |
| </p> |
| </td> |
| <td> |
| <p> |
| 1/λ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| 2 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 9 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| 6 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.exp_dist.references"></a><h5> |
| <a name="id1021881"></a> |
| <a class="link" href="exp_dist.html#math_toolkit.dist.dist_ref.dists.exp_dist.references">references</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| <a href="http://mathworld.wolfram.com/ExponentialDistribution.html" target="_top">Weisstein, |
| Eric W. "Exponential Distribution." From MathWorld--A Wolfram |
| Web Resource</a> |
| </li> |
| <li> |
| <a href="http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html" target="_top">Wolfram |
| Mathematica calculator</a> |
| </li> |
| <li> |
| <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm" target="_top">NIST |
| Exploratory Data Analysis</a> |
| </li> |
| <li> |
| <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Wikipedia |
| Exponential distribution</a> |
| </li> |
| </ul></div> |
| <p> |
| (See also the reference documentation for the related <a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme |
| Distributions</a>.) |
| </p> |
| <div class="itemizedlist"><ul type="disc"><li> |
| <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme |
| Value Distributions, Theory and Applications Samuel Kotz & Saralees |
| Nadarajah</a> discuss the relationship of the types of extreme |
| value distributions. |
| </li></ul></div> |
| </div> |
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| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
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