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<title>Exponential Distribution</title>
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.exp_dist"></a><a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>
</h4></div></div></div>
<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>
<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="Chapter 14. 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<></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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h0"></a>
<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
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="../../error_handling.html#math_toolkit.error_handling.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>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h1"></a>
<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
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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<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>,
<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>,
<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>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<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>,
<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>.
</p>
<p>
The domain of the random variable is [0, +∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h2"></a>
<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>
</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>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h3"></a>
<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>
</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>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h4"></a>
<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>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a href="http://mathworld.wolfram.com/ExponentialDistribution.html" target="_top">Weisstein,
Eric W. "Exponential Distribution." From MathWorld--A Wolfram
Web Resource</a>
</li>
<li class="listitem">
<a href="http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html" target="_top">Wolfram
Mathematica calculator</a>
</li>
<li class="listitem">
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm" target="_top">NIST
Exploratory Data Analysis</a>
</li>
<li class="listitem">
<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 class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<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>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>)
</p>
</div></td>
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