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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="id1029733"></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="id1029834"></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="id1029933"></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="id1029981"></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="id1030248"></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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