<title>Weibull Distribution</title>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</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">double</span><span class="special">,</span>
- <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 
19. 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> <a class="link" href="../../../policy.html" title="Chapter 
20. 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">weibull_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">weibull_distribution</span><span class="special"><></span> <span class="identifier">weibull</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 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">weibull_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
density function</a>:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">f(x; α, β) = (α/β) * (x / β)<sup>α - 1</sup> * e<sup>-(x/β)<sup>α</sup></sup></span>
+ </p></blockquote></div>
<p>
- f(x; α, β) = (α/β) * (x / β)<sup>α - 1</sup> * e<sup>-(x/β)<sup>α</sup></sup>
- </p>
-<p>
- For shape parameter α   > 0, and scale parameter β   > 0, and x > 0.
+ For shape parameter <span class="emphasis"><em>α</em></span> > 0, and scale parameter
+ <span class="emphasis"><em>β</em></span> > 0, and <span class="emphasis"><em>x</em></span> > 0.
</p>
<p>
The Weibull distribution is often used in the field of failure analysis;
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
- constant over time, then α   = 1, suggests that items are failing from
- random events.
+ constant over time, then <span class="emphasis"><em>α</em></span> = 1, suggests that items
+ are failing from random events.
</li>
<li class="listitem">
- decreases over time, then α   < 1, suggesting "infant mortality".
+ decreases over time, then <span class="emphasis"><em>α</em></span> < 1, suggesting
+ "infant mortality".
</li>
<li class="listitem">
- increases over time, then α   > 1, suggesting "wear out" -
- more likely to fail as time goes by.
+ increases over time, then <span class="emphasis"><em>α</em></span> > 1, suggesting
+ "wear out" - more likely to fail as time goes by.
</li>
</ul></div>
<p>
The following graph illustrates how the PDF varies with the shape parameter
- α:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf1.svg" align="middle"></span>
+ <span class="emphasis"><em>α</em></span>:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf1.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
While this graph illustrates how the PDF varies with the scale parameter
- β:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf2.svg" align="middle"></span>
+ <span class="emphasis"><em>β</em></span>:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf2.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.weibull_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.related_distributions"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.related_distributions">Related
distributions</a>
</h5>
<p>
- When
α   = 3, the <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
+ When
<span class="emphasis"><em>α</em></span> = 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>. When <span class="emphasis"><em>α</em></span> = 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
<span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.implementation"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.implementation">Implementation</a>
</h5>
<p>
- In the following table α   is the shape parameter of the distribution, β   is its
- scale parameter, <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>.
+ In the following table <span class="emphasis"><em>α</em></span> is the shape parameter of
+ the distribution, <span class="emphasis"><em>β</em></span> is its scale parameter, <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>
projects / platform / upstream / boost.git / blobdiff