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Subject: [Boost-commit] svn:boost r66748 - trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists
From: pbristow_at_[hidden]
Date: 2010-11-25 06:58:39

Author: pbristow
Date: 2010-11-25 06:58:25 EST (Thu, 25 Nov 2010)
New Revision: 66748
URL: http://svn.boost.org/trac/boost/changeset/66748

Log:
Changed file name from weibull.dist
Added:
trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html (contents, props changed)

Added: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html
==============================================================================
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+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html 2010-11-25 06:58:25 EST (Thu, 25 Nov 2010)
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+<html>
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
+<title>Weibull Distribution</title>
+<link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css">
+<meta name="generator" content="DocBook XSL Stylesheets V1.74.0">
+<link rel="home" href="../../../../index.html" title="Math Toolkit">
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+<table cellpadding="2" width="100%"><tr>
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+<td align="center">Home</td>
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+</div>
+<div class="section" lang="en">
+<div class="titlepage"><div><div><h5 class="title">
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist"></a><a class="link" href="weibull_dist.html" title="Weibull Distribution">Weibull
+ Distribution</a>
+</h5></div></div></div>
+
+
+
+<pre class="programlisting">#include <boost/math/distributions/weibull.hpp></pre>
+
+ 
+<pre class="programlisting">namespace boost{ namespace math{
+
+template <class RealType = double,
+ class <a class="link" href="../../../policy.html" title="Policies">Policy</a> = <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> >
+class weibull_distribution;
+
+typedef weibull_distribution<> weibull;
+
+template <class RealType, class <a class="link" href="../../../policy.html" title="Policies">Policy</a>>
+class weibull_distribution
+{
+public:
+ typedef RealType value_type;
+ typedef Policy policy_type;
+ // Construct:
+ weibull_distribution(RealType shape, RealType scale = 1)
+ // Accessors:
+ RealType shape()const;
+ RealType scale()const;
+};
+
+}} // namespaces
+</pre>
+
+ The <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
+ distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
+ density function</a>:
+ 
+
+ f(x; α, β) = (α/β) * (x / β)α - 1 * e-(x/β)α
+ 
+
+ For shape parameter α   > 0, and scale parameter β   > 0, and x > 0.
+ 
+
+ The Weibull distribution is often used in the field of failure analysis;
+ in particular it can mimic distributions where the failure rate varies
+ over time. If the failure rate is:
+ 
+<div class="itemizedlist"><ul type="disc">
+<li>
+ constant over time, then α   = 1, suggests that items are failing from
+ random events.
+ </li>
+<li>
+ decreases over time, then α   < 1, suggesting "infant mortality".
+ </li>
+<li>
+ increases over time, then α   > 1, suggesting "wear out"
+ - more likely to fail as time goes by.
+ </li>
+</ul></div>
+
+ The following graph illustrates how the PDF varies with the shape parameter
+ α:
+ 
+
+ <img src="../../../../../graphs/weibull_pdf1.png" align="middle">
+ 
+
+ While this graph illustrates how the PDF varies with the scale parameter
+ β:
+ 
+
+ <img src="../../../../../graphs/weibull_pdf2.png" align="middle">
+ 
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist.related_distributions"></a><h5>
+<a name="id1069340"></a>
+ <a class="link" href="weibull_dist.html#math_toolkit.dist.dist_ref.dists.weibull_dist.related_distributions">Related
+ distributions</a>
+ </h5>
+
+ When α   = 3, the <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
+ distribution</a> appears similar to the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">normal
+ distribution</a>. When α   = 1, the Weibull distribution reduces to the
+ <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential
+ distribution</a>. The relationship of the types of extreme value
+ distributions, of which the Weibull is but one, is discussed by <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme Value
+ Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah</a>.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist.member_functions"></a><h5>
+<a name="id1069379"></a>
+ <a class="link" href="weibull_dist.html#math_toolkit.dist.dist_ref.dists.weibull_dist.member_functions">Member
+ Functions</a>
+ </h5>
+<pre class="programlisting">weibull_distribution(RealType shape, RealType scale = 1);
+</pre>
+
+ Constructs a <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
+ distribution</a> with shape shape and scale
+ scale.
+ 
+
+ Requires that the shape and scale
+ parameters are both greater than zero, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
+ 
+<pre class="programlisting">RealType shape()const;
+</pre>
+
+ Returns the shape parameter of this distribution.
+ 
+<pre class="programlisting">RealType scale()const;
+</pre>
+
+ Returns the scale parameter of this distribution.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist.non_member_accessors"></a><h5>
+<a name="id1069535"></a>
+ <a class="link" href="weibull_dist.html#math_toolkit.dist.dist_ref.dists.weibull_dist.non_member_accessors">Non-member
+ Accessors</a>
+ </h5>
+
+ 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>.
+ 
+
+ The domain of the random variable is [0, ∞].
+ 
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist.accuracy"></a><h5>
+<a name="id1069632"></a>
+ <a class="link" href="weibull_dist.html#math_toolkit.dist.dist_ref.dists.weibull_dist.accuracy">Accuracy</a>
+ </h5>
+
+ The Weibull distribution is implemented in terms of the standard library
+ <code class="computeroutput">log</code> and <code class="computeroutput">exp</code> functions plus <a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a>
+ and <a class="link" href="../../../special/powers/log1p.html" title="log1p">log1p</a> and
+ as such should have very low error rates.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist.implementation"></a><h5>
+<a name="id1069675"></a>
+ <a class="link" href="weibull_dist.html#math_toolkit.dist.dist_ref.dists.weibull_dist.implementation">Implementation</a>
+ </h5>
+
+ In the following table α   is the shape parameter of the distribution, β   is
+ its scale parameter, x is the random variate, p
+ is the probability and q = 1-p.
+ 
+<div class="informaltable"><table class="table">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<thead><tr>
+<th>
+ 
+ Function
+ 
+ </th>
+<th>
+ 
+ Implementation Notes
+ 
+ </th>
+</tr></thead>
+<tbody>
+<tr>
+<td>
+ 
+ pdf
+ 
+ </td>
+<td>
+ 
+ Using the relation: pdf = αβ-α xα - 1 e-(x/beta)alpha
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf
+ 
+ </td>
+<td>
+ 
+ Using the relation: p = -<a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a>(-(x/β)α)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf complement
+ 
+ </td>
+<td>
+ 
+ Using the relation: q = e-(x/β)α
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile
+ 
+ </td>
+<td>
+ 
+ Using the relation: x = β * (-<a class="link" href="../../../special/powers/log1p.html" title="log1p">log1p</a>(-p))1/α
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile from the complement
+ 
+ </td>
+<td>
+ 
+ Using the relation: x = β * (-log(q))1/α
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mean
+ 
+ </td>
+<td>
+ 
+ β * Γ(1 + 1/α)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ variance
+ 
+ </td>
+<td>
+ 
+ β2(Γ(1 + 2/α) - Γ2(1 + 1/α))
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mode
+ 
+ </td>
+<td>
+ 
+ β((α - 1) / α)1/α
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ skewness
+ 
+ </td>
+<td>
+ 
+ Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
+ Eric W. "Weibull Distribution." From MathWorld--A
+ Wolfram Web Resource.</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis
+ 
+ </td>
+<td>
+ 
+ Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
+ Eric W. "Weibull Distribution." From MathWorld--A
+ Wolfram Web Resource.</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis excess
+ 
+ </td>
+<td>
+ 
+ Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
+ Eric W. "Weibull Distribution." From MathWorld--A
+ Wolfram Web Resource.</a>
+ 
+ </td>
+</tr>
+</tbody>
+</table></div>
+<a name="math_toolkit.dist.dist_ref.dists.weibull_dist.references"></a><h5>
+<a name="id1070007"></a>
+ <a class="link" href="weibull_dist.html#math_toolkit.dist.dist_ref.dists.weibull_dist.references">References</a>
+ </h5>
+<div class="itemizedlist"><ul type="disc">
+<li>
+ http://en.wikipedia.org/wiki/Weibull_distribution
+ </li>
+<li>
+ <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
+ Eric W. "Weibull Distribution." From MathWorld--A Wolfram
+ Web Resource.</a>
+ </li>
+<li>
+ <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm" target="_top">Weibull
+ in NIST Exploratory Data Analysis</a>
+ </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 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow,
+ Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and
+ Thijs van den Berg
+ Distributed under the Boost Software License, Version 1.0. (See accompanying
+ file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+ 
+</div></td>
+</tr></table>
+<hr>
+<div class="spirit-nav">
+<a accesskey="p" href="triangular_dist.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="uniform_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
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+</html>

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