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Subject: [Boost-commit] svn:boost r66750 - trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists
From: pbristow_at_[hidden]
Date: 2010-11-25 07:01:26

Author: pbristow
Date: 2010-11-25 07:01:15 EST (Thu, 25 Nov 2010)
New Revision: 66750
URL: http://svn.boost.org/trac/boost/changeset/66750

Log:
missing distribs
Added:
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html (contents, props changed)
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html (contents, props changed)
Removed:
   trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull.html

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@@ -0,0 +1,868 @@
+<html>
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
+<title>Geometric 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">
+<link rel="up" href="../dists.html" title="Distributions">
+<link rel="prev" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">
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+</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">Home</td>
+<td align="center">Libraries</td>
+<td align="center">People</td>
+<td align="center">FAQ</td>
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+<hr>
+<div class="spirit-nav">
+<a accesskey="p" href="gamma_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="inverse_chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+<div class="section" lang="en">
+<div class="titlepage"><div><div><h5 class="title">
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist"></a><a class="link" href="geometric_dist.html" title="Geometric Distribution">Geometric
+ Distribution</a>
+</h5></div></div></div>
+
+
+
+<pre class="programlisting">#include <boost/math/distributions/geometric.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 geometric_distribution;
+
+typedef geometric_distribution<> geometric;
+
+template <class RealType, class <a class="link" href="../../../policy.html" title="Policies">Policy</a>>
+class geometric_distribution
+{
+public:
+ typedef RealType value_type;
+ typedef Policy policy_type;
+ // Constructor from success_fraction:
+ geometric_distribution(RealType p);
+
+ // Parameter accessors:
+ RealType success_fraction() const;
+ RealType successes() const;
+
+ // Bounds on success fraction:
+ static RealType find_lower_bound_on_p(
+ RealType trials,
+ RealType successes,
+ RealType probability); // alpha
+ static RealType find_upper_bound_on_p(
+ RealType trials,
+ RealType successes,
+ RealType probability); // alpha
+
+ // Estimate min/max number of trials:
+ static RealType find_minimum_number_of_trials(
+ RealType k, // Number of failures.
+ RealType p, // Success fraction.
+ RealType probability); // Probability threshold alpha.
+ static RealType find_maximum_number_of_trials(
+ RealType k, // Number of failures.
+ RealType p, // Success fraction.
+ RealType probability); // Probability threshold alpha.
+};
+
+}} // namespaces
+</pre>
+
+ The class type <code class="computeroutput">geometric_distribution</code>
+ represents a <a href="http://en.wikipedia.org/wiki/geometric_distribution" target="_top">geometric
+ distribution</a>: it is used when there are exactly two mutually
+ exclusive outcomes of a <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli
+ trial</a>: these outcomes are labelled "success" and "failure".
+ 
+
+ For <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli
+ trials</a> each with success fraction p, the
+ geometric distribution gives the probability of observing k
+ trials (failures, events, occurrences, or arrivals) before the first
+ success.
+ 
+<div class="note"><table border="0" summary="Note">
+<tr>
+<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td>
+<th align="left">Note</th>
+</tr>
+<tr><td align="left" valign="top">
+ For this implementation, the set of trials includes
+ zero (unlike another definition where the set of trials
+ starts at one, sometimes named shifted).
+ </td></tr>
+</table></div>
+
+ The geometric distribution assumes that success_fraction p
+ is fixed for all k trials.
+ 
+
+ The probability that there are k failures before
+ the first success is
+ 
+
+    Pr(Y=k) = (1-p)kp
+ 
+
+ For example, when throwing a 6-face dice the success probability p
+ = 1/6 = 0.1666 ̇  . Throwing repeatedly until a three
+ appears, the probability distribution of the number of times not-a-three
+ is thrown is geometric.
+ 
+
+ Geometric distribution has the Probability Density Function PDF:
+ 
+
+    (1-p)kp
+ 
+
+ The following graph illustrates how the PDF and CDF vary for three examples
+ of the success fraction p, (when considering the
+ geometric distribution as a continuous function),
+ 
+
+ <img src="../../../../../graphs/geometric_pdf_2.png" align="middle">
+ 
+
+ <img src="../../../../../graphs/geometric_cdf_2.png" align="middle">
+ 
+
+ and as discrete.
+ 
+
+ <img src="../../../../../graphs/geometric_pdf_discrete.png" align="middle">
+ 
+
+ <img src="../../../../../graphs/geometric_cdf_discrete.png" align="middle">
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.related_distributions"></a><h5>
+<a name="id1027789"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.related_distributions">Related
+ Distributions</a>
+ </h5>
+
+ The geometric distribution is a special case of the <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a> with successes parameter r
+ = 1, so only one first and only success is required : thus by definition
+    <code class="computeroutput">geometric(p) == negative_binomial(1,
+ p)</code>
+ 
+<pre class="programlisting">negative_binomial_distribution(RealType r, RealType success_fraction);
+negative_binomial nb(1, success_fraction);
+geometric g(success_fraction);
+ASSERT(pdf(nb, 1) == pdf(g, 1));
+</pre>
+
+ This implementation uses real numbers for the computation throughout
+ (because it uses the real-valued power
+ and exponential functions). So to obtain a conventional strictly-discrete
+ geometric distribution you must ensure that an integer value is provided
+ for the number of trials (random variable) k, and
+ take integer values (floor or ceil functions) from functions that return
+ a number of successes.
+ 
+<div class="caution"><table border="0" summary="Caution">
+<tr>
+<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../../doc/src/images/caution.png"></td>
+<th align="left">Caution</th>
+</tr>
+<tr><td align="left" valign="top">
+
+ The geometric distribution is a discrete distribution: internally,
+ functions like the <code class="computeroutput">cdf</code>
+ and <code class="computeroutput">pdf</code> are treated
+ "as if" they are continuous functions, but in reality the
+ results returned from these functions only have meaning if an integer
+ value is provided for the random variate argument.
+ 
+
+ The quantile function will by default return an integer result that
+ has been rounded outwards. That is to say lower
+ quantiles (where the probability is less than 0.5) are rounded downward,
+ and upper quantiles (where the probability is greater than 0.5) are
+ rounded upwards. This behaviour ensures that if an X% quantile is requested,
+ then at least the requested coverage will be present
+ in the central region, and no more than the requested
+ coverage will be present in the tails.
+ 
+
+ This behaviour can be changed so that the quantile functions are rounded
+ differently, or even return a real-valued result using <a class="link" href="../../../policy/pol_overview.html" title="Policy Overview">Policies</a>.
+ It is strongly recommended that you read the tutorial <a class="link" href="../../../policy/pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
+ Quantiles of Discrete Distributions</a> before using the quantile
+ function on the geometric distribution. The <a class="link" href="../../../policy/pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
+ docs</a> describe how to change the rounding policy for these distributions.
+ 
+</td></tr>
+</table></div>
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.member_functions"></a><h5>
+<a name="id1028074"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.member_functions">Member
+ Functions</a>
+ </h5>
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.constructor"></a><h6>
+<a name="id1028090"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.constructor">Constructor</a>
+ </h6>
+<pre class="programlisting">geometric_distribution(RealType p);
+</pre>
+
+ Constructor: p or success_fraction is the probability
+ of success of a single trial.
+ 
+
+ Requires: <code class="computeroutput">0 <=
+ p <=
+ 1</code>.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.accessors"></a><h6>
+<a name="id1028169"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.accessors">Accessors</a>
+ </h6>
+<pre class="programlisting">RealType success_fraction() const; // successes / trials (0 <= p <= 1)
+</pre>
+
+ Returns the success_fraction parameter p from which
+ this distribution was constructed.
+ 
+<pre class="programlisting">RealType successes() const; // required successes always one,
+// included for compatibility with negative binomial distribution
+// with successes r == 1.
+</pre>
+
+ Returns unity.
+ 
+
+ The following functions are equivalent to those provided for the negative
+ binomial, with successes = 1, but are provided here for completeness.
+ 
+
+ The best method of calculation for the following functions is disputed:
+ see <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
+ Distribution</a> and <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a> for more discussion.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.lower_bound_on_success_fraction_parameter__emphasis_p__emphasis_"></a><h6>
+<a name="id1028277"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.lower_bound_on_success_fraction_parameter__emphasis_p__emphasis_">Lower
+ Bound on success_fraction Parameter p</a>
+ </h6>
+<pre class="programlisting">static RealType find_lower_bound_on_p(
+ RealType failures,
+ RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
+</pre>
+
+ Returns a lower bound on the success
+ fraction:
+ 
+<div class="variablelist">
+
+<dl>
+<dt>failures</dt>
+<dd>
+ The total number of failures before the 1st success.
+ </dd>
+<dt>alpha</dt>
+<dd>
+ The largest acceptable probability that the true value of the success
+ fraction is less than the value
+ returned.
+ </dd>
+</dl>
+</div>
+
+ For example, if you observe k failures from n
+ trials the best estimate for the success fraction is simply 1/n,
+ but if you want to be 95% sure that the true value is greater
+ than some value, pmin, then:
+ 
+<pre class="programlisting">pmin = geometric_distribution<RealType>::
+ find_lower_bound_on_p(failures, 0.05);
+</pre>
+
+ <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See
+ negative_binomial confidence interval example.</a>
+ 
+
+ This function uses the Clopper-Pearson method of computing the lower
+ bound on the success fraction, whilst many texts refer to this method
+ as giving an "exact" result in practice it produces an interval
+ that guarantees at least the coverage required,
+ and may produce pessimistic estimates for some combinations of failures
+ and successes. See:
+ 
+
+ <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong
+ Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some
+ Discrete Distributions. Computational statistics and data analysis, 2005,
+ vol. 48, no3, 605-621</a>.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.upper_bound_on_success_fraction_parameter_p"></a><h6>
+<a name="id1028507"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.upper_bound_on_success_fraction_parameter_p">Upper
+ Bound on success_fraction Parameter p</a>
+ </h6>
+<pre class="programlisting">static RealType find_upper_bound_on_p(
+ RealType trials,
+ RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
+</pre>
+
+ Returns an upper bound on the success
+ fraction:
+ 
+<div class="variablelist">
+
+<dl>
+<dt>trials</dt>
+<dd>
+ The total number of trials conducted.
+ </dd>
+<dt>alpha</dt>
+<dd>
+ The largest acceptable probability that the true value of the success
+ fraction is greater than the value
+ returned.
+ </dd>
+</dl>
+</div>
+
+ For example, if you observe k successes from n
+ trials the best estimate for the success fraction is simply k/n,
+ but if you want to be 95% sure that the true value is less
+ than some value, pmax, then:
+ 
+<pre class="programlisting">pmax = geometric_distribution<RealType>::find_upper_bound_on_p(
+ k, 0.05);
+</pre>
+
+ <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See
+ negative binomial confidence interval example.</a>
+ 
+
+ This function uses the Clopper-Pearson method of computing the lower
+ bound on the success fraction, whilst many texts refer to this method
+ as giving an "exact" result in practice it produces an interval
+ that guarantees at least the coverage required,
+ and may produce pessimistic estimates for some combinations of failures
+ and successes. See:
+ 
+
+ <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong
+ Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some
+ Discrete Distributions. Computational statistics and data analysis, 2005,
+ vol. 48, no3, 605-621</a>.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_at_least_a_certain_number_of_failures"></a><h6>
+<a name="id1028735"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_at_least_a_certain_number_of_failures">Estimating
+ Number of Trials to Ensure at Least a Certain Number of Failures</a>
+ </h6>
+<pre class="programlisting">static RealType find_minimum_number_of_trials(
+ RealType k, // number of failures.
+ RealType p, // success fraction.
+ RealType alpha); // probability threshold (0.05 equivalent to 95%).
+</pre>
+
+ This functions estimates the number of trials required to achieve a certain
+ probability that more than k
+ failures will be observed.
+ 
+<div class="variablelist">
+
+<dl>
+<dt>k</dt>
+<dd>
+ The target number of failures to be observed.
+ </dd>
+<dt>p</dt>
+<dd>
+ The probability of success for each trial.
+ </dd>
+<dt>alpha</dt>
+<dd>
+ The maximum acceptable risk that only k
+ failures or fewer will be observed.
+ </dd>
+</dl>
+</div>
+
+ For example:
+ 
+<pre class="programlisting">geometric_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
+</pre>
+
+ Returns the smallest number of trials we must conduct to be 95% (1-0.05)
+ sure of seeing 10 failures that occur with frequency one half.
+ 
+
+ <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial.">Worked
+ Example.</a>
+ 
+
+ This function uses numeric inversion of the geometric distribution to
+ obtain the result: another interpretation of the result is that it finds
+ the number of trials (failures) that will lead to an alpha
+ probability of observing k failures or fewer.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_a_maximum_number_of_failures_or_less"></a><h6>
+<a name="id1028974"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_a_maximum_number_of_failures_or_less">Estimating
+ Number of Trials to Ensure a Maximum Number of Failures or Less</a>
+ </h6>
+<pre class="programlisting">static RealType find_maximum_number_of_trials(
+ RealType k, // number of failures.
+ RealType p, // success fraction.
+ RealType alpha); // probability threshold (0.05 equivalent to 95%).
+</pre>
+
+ This functions estimates the maximum number of trials we can conduct
+ and achieve a certain probability that k failures
+ or fewer will be observed.
+ 
+<div class="variablelist">
+
+<dl>
+<dt>k</dt>
+<dd>
+ The maximum number of failures to be observed.
+ </dd>
+<dt>p</dt>
+<dd>
+ The probability of success for each trial.
+ </dd>
+<dt>alpha</dt>
+<dd>
+ The maximum acceptable risk that more than
+ k failures will be observed.
+ </dd>
+</dl>
+</div>
+
+ For example:
+ 
+<pre class="programlisting">geometric_distribution<RealType>::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05);
+</pre>
+
+ Returns the largest number of trials we can conduct and still be 95%
+ sure of seeing no failures that occur with frequency one in one million.
+ 
+
+ This function uses numeric inversion of the geometric distribution to
+ obtain the result: another interpretation of the result, is that it finds
+ the number of trials that will lead to an alpha
+ probability of observing more than k failures.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.non_member_accessors"></a><h5>
+<a name="id1029209"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_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>.
+ 
+
+ However it's worth taking a moment to define what these actually mean
+ in the context of this distribution:
+ 
+<div class="table">
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.meaning_of_the_non_member_accessors_"></a>Table 14. Meaning of the non-member accessors.
+<div class="table-contents"><table class="table" summary="Meaning of the non-member accessors.">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<thead><tr>
+<th>
+ 
+ Function
+ 
+ </th>
+<th>
+ 
+ Meaning
+ 
+ </th>
+</tr></thead>
+<tbody>
+<tr>
+<td>
+ 
+ <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>
+ 
+ </td>
+<td>
+ 
+ The probability of obtaining exactly
+ k failures from k trials with
+ success fraction p. For example:
+ 
+ 
+
+
+<pre class="programlisting">pdf(geometric(p), k)</pre>
+
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>
+ 
+ </td>
+<td>
+ 
+ The probability of obtaining k failures
+ or fewer from k trials with
+ success fraction p and success on the last trial. For example:
+ 
+ 
+
+
+<pre class="programlisting">cdf(geometric(p), k)</pre>
+
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <a class="link" href="../nmp.html#math.dist.ccdf">Complement of the Cumulative
+ Distribution Function</a>
+ 
+ </td>
+<td>
+ 
+ The probability of obtaining more than
+ k failures from k trials with
+ success fraction p and success on the last trial. For example:
+ 
+ 
+
+
+<pre class="programlisting">cdf(complement(geometric(p), k))</pre>
+
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>
+ 
+ </td>
+<td>
+ 
+ The greatest number of failures
+ k expected to be observed from k
+ trials with success fraction p, at probability
+ P. Note that the value returned is a real-number,
+ and not an integer. Depending on the use case you may want
+ to take either the floor or ceiling of the real result. For
+ example:
+
+<pre class="programlisting">quantile(geometric(p), P)</pre>
+
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <a class="link" href="../nmp.html#math.dist.quantile_c">Quantile from the complement
+ of the probability</a>
+ 
+ </td>
+<td>
+ 
+ The smallest number of failures
+ k expected to be observed from k
+ trials with success fraction p, at probability
+ P. Note that the value returned is a real-number,
+ and not an integer. Depending on the use case you may want
+ to take either the floor or ceiling of the real result. For
+ example:
+
+<pre class="programlisting">quantile(complement(geometric(p), P))</pre>
+
+ 
+ </td>
+</tr>
+</tbody>
+</table></div>
+</div>
+ <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.accuracy"></a><h5>
+<a name="id1029719"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.accuracy">Accuracy</a>
+ </h5>
+
+ This distribution is implemented using the pow and exp functions, so
+ most results are accurate within a few epsilon for the RealType. For
+ extreme values of <code class="computeroutput">double</code>
+ p, for example 0.9999999999, accuracy can fall significantly,
+ for example to 10 decimal digits (from 16).
+ 
+<a name="math_toolkit.dist.dist_ref.dists.geometric_dist.implementation"></a><h5>
+<a name="id1029752"></a>
+ <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.implementation">Implementation</a>
+ </h5>
+
+ In the following table, p is the probability that
+ any one trial will be successful (the success fraction), k
+ is the number of failures, p is the probability
+ and q = 1-p, x is the given
+ probability to estimate the expected number of failures using the quantile.
+ 
+<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>
+ 
+ pdf = p * pow(q, k)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf
+ 
+ </td>
+<td>
+ 
+ cdf = 1 - qk=1
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf complement
+ 
+ </td>
+<td>
+ 
+ exp(log1p(-p) * (k+1))
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile
+ 
+ </td>
+<td>
+ 
+ k = log1p(-x) / log1p(-p) -1
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile from the complement
+ 
+ </td>
+<td>
+ 
+ k = log(x) / log1p(-p) -1
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mean
+ 
+ </td>
+<td>
+ 
+ (1-p)/p
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ variance
+ 
+ </td>
+<td>
+ 
+ (1-p)/p²
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mode
+ 
+ </td>
+<td>
+ 
+ 0
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ skewness
+ 
+ </td>
+<td>
+ 
+ (2-p)/√q
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis
+ 
+ </td>
+<td>
+ 
+ 9+p²/q
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis excess
+ 
+ </td>
+<td>
+ 
+ 6 +p²/q
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ parameter estimation member functions
+ 
+ </td>
+<td>
+ 
+ See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <code class="computeroutput">find_lower_bound_on_p</code>
+ 
+ </td>
+<td>
+ 
+ See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <code class="computeroutput">find_upper_bound_on_p</code>
+ 
+ </td>
+<td>
+ 
+ See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <code class="computeroutput">find_minimum_number_of_trials</code>
+ 
+ </td>
+<td>
+ 
+ See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ <code class="computeroutput">find_maximum_number_of_trials</code>
+ 
+ </td>
+<td>
+ 
+ See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative
+ Binomial Distribution</a>
+ 
+ </td>
+</tr>
+</tbody>
+</table></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="gamma_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="inverse_chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+</body>
+</html>

Added: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html
==============================================================================
--- (empty file)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html 2010-11-25 07:01:15 EST (Thu, 25 Nov 2010)
@@ -0,0 +1,454 @@
+<html>
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
+<title>Inverse Gaussian (or Inverse Normal) 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">
+<link rel="up" href="../dists.html" title="Distributions">
+<link rel="prev" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">
+<link rel="next" href="hypergeometric_dist.html" title="Hypergeometric Distribution">
+</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">Home</td>
+<td align="center">Libraries</td>
+<td align="center">People</td>
+<td align="center">FAQ</td>
+<td align="center">More</td>
+</tr></table>
+<hr>
+<div class="spirit-nav">
+<a accesskey="p" href="inverse_gamma_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="hypergeometric_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+<div class="section" lang="en">
+<div class="titlepage"><div><div><h5 class="title">
+<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist"></a><a class="link" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">Inverse
+ Gaussian (or Inverse Normal) Distribution</a>
+</h5></div></div></div>
+
+
+
+<pre class="programlisting">#include <boost/math/distributions/wald.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 wald_distribution
+{
+public:
+ typedef RealType value_type;
+ typedef Policy policy_type;
+
+ wald_distribution(RealType mean = 1, RealType scale = 1);
+
+ RealType mean()const; // mean default 1.
+ RealType scale()const; // Optional scale, default 1 (unscaled).
+ RealType scale()const; // Shape = scale/mean.
+};
+typedef inverse_gaussian_distribution<double> inverse_gaussian;
+
+}} // namespace boost // namespace math
+</pre>
+
+ The Inverse Gaussian distribution distribution is a continuous probability
+ distribution.
+ 
+
+ The distribution is also called 'normal-inverse Gaussian distribution',
+ and 'normal Inverse' distribution.
+ 
+
+ It is also convenient to provide unity as default for both mean and scale.
+ This is the Standard form for all distributions. The Inverse Gaussian
+ distribution was first studied in relation to Brownian motion. In 1956
+ M.C.K. Tweedie used the name Inverse Gaussian because there is an inverse
+ relationship between the time to cover a unit distance and distance covered
+ in unit time. The inverse Gaussian is one of family of distributions
+ that have been called the <a href="http://en.wikipedia.org/wiki/Tweedie_distributions" target="_top">Tweedie
+ distributions</a>.
+ 
+
+ (So inverse in the name may mislead: it does not relate to the inverse of a distribution).
+ 
+
+ The tails of the distribution decrease more slowly than the normal distribution.
+ It is therefore suitable to model phenomena where numerically large values
+ are more probable than is the case for the normal distribution. For stock
+ market returns and prices, a key characteristic is that it models that
+ extremely large variations from typical (crashes) can occur even when
+ almost all (normal) variations are small.
+ 
+
+ Examples are returns from financial assets and turbulent wind speeds.
+ 
+
+ The normal-inverse Gaussian distributions form a subclass of the generalised
+ hyperbolic distributions.
+ 
+
+ See distribution.
+ <a href="http://mathworld.wolfram.com/InverseGaussianDistribution.html" target="_top">Weisstein,
+ Eric W. "Inverse Gaussian Distribution." From MathWorld--A
+ Wolfram Web Resource.</a>
+ 
+
+ If you want a <code class="computeroutput">double</code> precision
+ inverse_gaussian distribution you can use
+ 
+
+
+
+<pre class="programlisting">boost::math::inverse_gaussian_distribution<></pre>
+
+ 
+
+ or, more conveniently, you can write
+ 
+<pre class="programlisting">using boost::math::inverse_gaussian;
+inverse_gaussian my_ig(2, 3);
+</pre>
+
+ For mean parameters μ and scale (also called precision) parameter λ, and
+ random variate x, the inverse_gaussian distribution is defined by the
+ probability density function (PDF):
+ 
+
+    f(x;μ, λ) = √(λ/2πx3) e-λ(x-μ)²/2μ²x
+ 
+
+ and Cumulative Density Function (CDF):
+ 
+
+    F(x;μ, λ) = Φ{√(λx) (xμ-1)} + e2μ/λ Φ{-√(λ/μ) (1+x/μ)}
+ 
+
+ where Φ is the standard normal distribution CDF.
+ 
+
+ The following graphs illustrate how the PDF and CDF of the inverse_gaussian
+ distribution varies for a few values of parameters μ and λ:
+ 
+
+ <img src="../../../../../graphs/inverse_gaussian_pdf.png" align="middle">
+ 
+
+ <img src="../../../../../graphs/inverse_gaussian_cdf.png" align="middle">
+ 
+
+ Tweedie also provided 3 other parameterisations where (μ and λ) are replaced
+ by their ratio φ = λ/μ and by 1/μ: these forms may be more suitable for Bayesian
+ applications. These can be found on Seshadri, page 2 and are also discussed
+ by Chhikara and Folks on page 105. Another related parameterisation,
+ the __wald_distrib (where mean μ is unity) is also provided.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.member_functions"></a><h5>
+<a name="id1034890"></a>
+ <a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.member_functions">Member
+ Functions</a>
+ </h5>
+<pre class="programlisting">inverse_gaussian_distribution(RealType df = 1, RealType scale = 1); // optionally scaled.
+</pre>
+
+ Constructs an inverse_gaussian distribution with μ mean, and scale λ, with
+ both default values 1.
+ 
+
+ Requires that both the mean μ parameter and scale λ are greater than zero,
+ otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
+ 
+<pre class="programlisting">RealType mean()const;
+</pre>
+
+ Returns the mean μ parameter of this distribution.
+ 
+<pre class="programlisting">RealType scale()const;
+</pre>
+
+ Returns the scale λ parameter of this distribution.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.non_member_accessors"></a><h5>
+<a name="id1035030"></a>
+ <a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_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 variate is [0,+∞).
+ 
+<div class="note"><table border="0" summary="Note">
+<tr>
+<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td>
+<th align="left">Note</th>
+</tr>
+<tr><td align="left" valign="top">
+ Unlike some definitions, this implementation supports a random variate
+ equal to zero as a special case, returning zero for both pdf and cdf.
+ </td></tr>
+</table></div>
+<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.accuracy"></a><h5>
+<a name="id1035137"></a>
+ <a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.accuracy">Accuracy</a>
+ </h5>
+
+ The inverse_gaussian distribution is implemented in terms of the exponential
+ function and standard normal distribution N0,1 Φ :
+ refer to the accuracy data for those functions for more information.
+ But in general, gamma (and thus inverse gamma) results are often accurate
+ to a few epsilon, >14 decimal digits accuracy for 64-bit double.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.implementation"></a><h5>
+<a name="id1035160"></a>
+ <a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.implementation">Implementation</a>
+ </h5>
+
+ In the following table μ is the mean parameter and λ is the scale parameter
+ of the inverse_gaussian distribution, x is the random
+ variate, p is the probability and q =
+ 1-p its complement. Parameters μ for shape and λ for scale are
+ used for the inverse gaussian function.
+ 
+<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>
+ 
+ √(λ/ 2πx3) e-λ(x - μ)²/ 2μ²x
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf
+ 
+ </td>
+<td>
+ 
+ Φ{√(λx) (xμ-1)} + e2μ/λ Φ{-√(λ/μ) (1+x/μ)}
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf complement
+ 
+ </td>
+<td>
+ 
+ using complement of Φ above.
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile
+ 
+ </td>
+<td>
+ 
+ No closed form known. Estimated using a guess refined by Newton-Raphson
+ iteration.
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile from the complement
+ 
+ </td>
+<td>
+ 
+ No closed form known. Estimated using a guess refined by Newton-Raphson
+ iteration.
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mode
+ 
+ </td>
+<td>
+ 
+ μ {√(1+9μ²/4λ²)² - 3μ/2λ}
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ median
+ 
+ </td>
+<td>
+ 
+ no closed form analytic equation is known, but is evaluated
+ as quantile(0.5)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mean
+ 
+ </td>
+<td>
+ 
+ μ
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ variance
+ 
+ </td>
+<td>
+ 
+ μ³/λ
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ skewness
+ 
+ </td>
+<td>
+ 
+ 3 √ (μ/λ)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis_excess
+ 
+ </td>
+<td>
+ 
+ 15μ/λ
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis
+ 
+ </td>
+<td>
+ 
+ 12μ/λ
+ 
+ </td>
+</tr>
+</tbody>
+</table></div>
+<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.references"></a><h5>
+<a name="id1035455"></a>
+ <a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.references">References</a>
+ </h5>
+<div class="orderedlist"><ol type="1">
+<li>
+ Wald, A. (1947). Sequential analysis. Wiley, NY.
+ </li>
+<li>
+ The Tnverse Gaussian distribution : theory, methodology, and applications,
+ Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
+ </li>
+<li>
+ The Inverse Gaussian distribution : statistical theory and applications,
+ Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
+ </li>
+<li>
+ <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy
+ and Scipy Documentation</a>.
+ </li>
+<li>
+ <a href="http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html" target="_top">R
+ statmod invgauss functions</a>.
+ </li>
+<li>
+ <a href="http://cran.r-project.org/web/packages/SuppDists/index.html" target="_top">R
+ SuppDists invGauss functions</a>. (Note that these R implementations
+ names differ in case).
+ </li>
+<li>
+ <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss
+ help</a>.
+ </li>
+<li>
+ <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss
+ R source</a>.
+ </li>
+<li>
+ <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald,
+ qwald</a>.
+ </li>
+<li>
+ <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton
+ Webs wald</a>.
+ </li>
+</ol></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="inverse_gamma_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="hypergeometric_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+</body>
+</html>

Deleted: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull.html
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull.html 2010-11-25 07:01:15 EST (Thu, 25 Nov 2010)
+++ (empty file)
@@ -1,373 +0,0 @@
-<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">
-<link rel="up" href="../dists.html" title="Distributions">
-<link rel="prev" href="triangular_dist.html" title="Triangular Distribution">
-<link rel="next" href="uniform_dist.html" title="Uniform Distribution">
-</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">Home</td>
-<td align="center">Libraries</td>
-<td align="center">People</td>
-<td align="center">FAQ</td>
-<td align="center">More</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>
-<div class="section" lang="en">
-<div class="titlepage"><div><div><h5 class="title">
-<a name="math_toolkit.dist.dist_ref.dists.weibull"></a><a class="link" href="weibull.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.related_distributions"></a><h5>
-<a name="id1156987"></a>
- <a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.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.member_functions"></a><h5>
-<a name="id1157028"></a>
- <a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.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.non_member_accessors"></a><h5>
-<a name="id1157217"></a>
- <a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.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.accuracy"></a><h5>
-<a name="id1157317"></a>
- <a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.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.implementation"></a><h5>
-<a name="id1157357"></a>
- <a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.implementation">Implementation</a>
- </h5>
-
- In the following table α is the shape parameter of the distribution, β is
- it's 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.references"></a><h5>
-<a name="id1157685"></a>
- <a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.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>
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