mirror of
https://github.com/boostorg/math.git
synced 2026-01-22 17:32:13 +00:00
330 lines
16 KiB
HTML
330 lines
16 KiB
HTML
<html>
|
||
<head>
|
||
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
||
<title>Exponential Distribution</title>
|
||
<link rel="stylesheet" href="../../../math.css" type="text/css">
|
||
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
||
<link rel="home" href="../../../index.html" title="Math Toolkit 4.2.1">
|
||
<link rel="up" href="../dists.html" title="Distributions">
|
||
<link rel="prev" href="empirical_cdf.html" title="Empirical Cumulative Distribution Function">
|
||
<link rel="next" href="extreme_dist.html" title="Extreme Value Distribution">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||
</head>
|
||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||
<table cellpadding="2" width="100%"><tr>
|
||
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
|
||
<td align="center"><a href="../../../../../../../index.html">Home</a></td>
|
||
<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
|
||
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
||
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
||
<td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
|
||
</tr></table>
|
||
<hr>
|
||
<div class="spirit-nav">
|
||
<a accesskey="p" href="empirical_cdf.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="extreme_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
</div>
|
||
<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 22. 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 22. 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>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="inlinemediaobject"><img src="../../../../equations/exponential_dist_ref1.svg"></span>
|
||
|
||
</p></blockquote></div>
|
||
<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>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="inlinemediaobject"><img src="../../../../graphs/exponential_pdf.svg" align="middle"></span>
|
||
|
||
</p></blockquote></div>
|
||
<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="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&gbpv=0" 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>
|
||
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
|
||
<hr>
|
||
<div class="spirit-nav">
|
||
<a accesskey="p" href="empirical_cdf.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="extreme_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
</div>
|
||
</body>
|
||
</html>
|