<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Beta 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="bernoulli_dist.html" title="Bernoulli Distribution">
<link rel="next" href="binomial_dist.html" title="Binomial 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"><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="bernoulli_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="binomial_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.beta_dist"></a><a class="link" href="beta_dist.html" title="Beta Distribution"> Beta
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">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
<p>
</p>
<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="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">beta_distribution</span><span class="special">;</span>
<span class="comment">// typedef beta_distribution<double> beta;
</span><span class="comment">// Note that this is deliberately NOT provided,
</span><span class="comment">// to avoid a clash with the function name beta.
</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">beta_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="comment">// Constructor from two shape parameters, alpha & beta:
</span> <span class="identifier">beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">);</span>
<span class="comment">// Parameter accessors:
</span> <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="comment">// Parameter estimators of alpha or beta from mean and variance.
</span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.
</span> <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.
</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.
</span> <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.
</span>
<span class="comment">// Parameter estimators from from
</span> <span class="comment">// either alpha or beta, and x and probability.
</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.
</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// x.
</span> <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// cdf
</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.
</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.
</span> <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.
</span><span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces
</span></pre>
<p>
The class type <code class="computeroutput"><span class="identifier">beta_distribution</span></code>
represents a <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
</a> <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
distribution function</a>.
</p>
<p>
The <a href="http://mathworld.wolfram.com/BetaDistribution.htm" target="_top">beta
distribution </a> is used as a <a href="http://en.wikipedia.org/wiki/Prior_distribution" target="_top">prior
distribution</a> for binomial proportions in <a href="http://mathworld.wolfram.com/BayesianAnalysis.html" target="_top">Bayesian
analysis</a>.
</p>
<p>
See also: <a href="http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html" target="_top">beta
distribution</a> and <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
statistics</a>.
</p>
<p>
How the beta distribution is used for <a href="http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt" target="_top">Bayesian
analysis of one parameter models</a> is discussed by Jeff Grynaviski.
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
distribution</a> defined on the interval [0,1] is given by:
</p>
<p>
f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)
</p>
<p>
where B(α, β) is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
function</a>, implemented in this library as <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta</a>.
Division by the beta function ensures that the pdf is normalized to the
range zero to unity.
</p>
<p>
The following graph illustrates examples of the pdf for various values
of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and
might be approximated by a symmetrical triangular distribution.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/beta_pdf.png" align="middle"></span>
</p>
<p>
If α = β = 1, then it is a ​
<a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
distribution</a>, equal to unity in the entire interval x = 0 to
1. If α ​ and β ​ are < 1, then the pdf is U-shaped. If α != β, then the shape
is asymmetric and could be approximated by a triangle whose apex is away
from the centre (where x = half).
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.member_functions"></a><h5>
<a name="id1017857"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.member_functions">Member
Functions</a>
</h5>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.constructor"></a><h6>
<a name="id1017873"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.constructor">Constructor</a>
</h6>
<pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">);</span>
</pre>
<p>
Constructs a beta distribution with shape parameters <span class="emphasis"><em>alpha</em></span>
and <span class="emphasis"><em>beta</em></span>.
</p>
<p>
Requires alpha,beta > 0,otherwise <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
is called. Note that technically the beta distribution is defined for
alpha,beta >= 0, but it's not clear whether any program can actually
make use of that latitude or how many of the non-member functions can
be usefully defined in that case. Therefore for now, we regard it as
an error if alpha or beta is zero.
</p>
<p>
For example:
</p>
<pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special"><></span> <span class="identifier">mybeta</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span>
</pre>
<p>
Constructs a the beta distribution with alpha=2 and beta=5 (shown in
yellow in the graph above).
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.parameter_accessors"></a><h6>
<a name="id1017988"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.parameter_accessors">Parameter
Accessors</a>
</h6>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the parameter <span class="emphasis"><em>alpha</em></span> from which this distribution
was constructed.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the parameter <span class="emphasis"><em>beta</em></span> from which this distribution
was constructed.
</p>
<p>
So for example:
</p>
<pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special"><></span> <span class="identifier">mybeta</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span>
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">mybeta</span><span class="special">.</span><span class="identifier">alpha</span><span class="special">()</span> <span class="special">==</span> <span class="number">2.</span><span class="special">);</span> <span class="comment">// mybeta.alpha() returns 2
</span><span class="identifier">assert</span><span class="special">(</span><span class="identifier">mybeta</span><span class="special">.</span><span class="identifier">beta</span><span class="special">()</span> <span class="special">==</span> <span class="number">5.</span><span class="special">);</span> <span class="comment">// mybeta.beta() returns 5
</span></pre>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.parameter_estimators"></a><h5>
<a name="id1018189"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.parameter_estimators">Parameter
Estimators</a>
</h5>
<p>
Two pairs of parameter estimators are provided.
</p>
<p>
One estimates either α ​ or β ​
from presumed-known mean and variance.
</p>
<p>
The other pair estimates either α ​ or β ​ from the cdf and x.
</p>
<p>
It is also possible to estimate α ​ and β ​ from 'known' mode & quantile.
For example, calculators are provided by the <a href="http://www.ausvet.com.au/pprev/content.php?page=PPscript" target="_top">Pooled
Prevalence Calculator</a> and <a href="http://www.epi.ucdavis.edu/diagnostictests/betabuster.html" target="_top">Beta
Buster</a> but this is not yet implemented here.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.
</span> <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.
</span></pre>
<p>
Returns the unique value of α that corresponds to a beta distribution with
mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.
</span> <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.
</span></pre>
<p>
Returns the unique value of β that corresponds to a beta distribution with
mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.
</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// x.
</span> <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf
</span></pre>
<p>
Returns the value of α that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
</p>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.
</span> <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.
</span> <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.
</span></pre>
<p>
Returns the value of β that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.non_member_accessor_functions"></a><h5>
<a name="id1018639"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.non_member_accessor_functions">Non-member
Accessor Functions</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 formulae for calculating these are shown in the table below, and
at <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
Mathworld</a>.
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.applications"></a><h5>
<a name="id1018741"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.applications">Applications</a>
</h5>
<p>
The beta distribution can be used to model events constrained to take
place within an interval defined by a minimum and maximum value: so it
is used in project management systems.
</p>
<p>
It is also widely used in <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian
statistical inference</a>.
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.related_distributions"></a><h5>
<a name="id1018767"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.related_distributions">Related
distributions</a>
</h5>
<p>
The beta distribution with both α ​ and β = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
distribution</a>.
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular</a>
is used when less precise information is available.
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
distribution</a> is closely related when α ​ and β ​ are integers.
</p>
<p>
With integer values of α ​ and β ​ the distribution B(i, j) is that of the j-th
highest of a sample of i + j + 1 independent random variables uniformly
distributed between 0 and 1. The cumulative probability from 0 to x is
thus the probability that the j-th highest value is less than x. Or it
is the probability that that at least i of the random variables are less
than x, a probability given by summing over the <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
Distribution</a> with its p parameter set to x.
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.accuracy"></a><h5>
<a name="id1018813"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.accuracy">Accuracy</a>
</h5>
<p>
This distribution is implemented using the <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta
functions</a> <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta</a>
and <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
beta functions</a> <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
and <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>;
please refer to these functions for information on accuracy.
</p>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.implementation"></a><h5>
<a name="id1018858"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.implementation">Implementation</a>
</h5>
<p>
In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
are the parameters α and β, <span class="emphasis"><em>x</em></span> is the random variable,
<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>
f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)
</p>
<p>
Implemented using <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(a,
b, x).
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<p>
Using the incomplete beta function <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(a,
b, x)
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf complement
</p>
</td>
<td>
<p>
<a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(a,
b, x)
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Using the inverse incomplete beta function <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(a,
b, p)
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
<a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(a,
b, q)
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">a</span><span class="special">/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
<span class="identifier">b</span> <span class="special">/</span>
<span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)^</span><span class="number">2</span>
<span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span>
<span class="identifier">b</span> <span class="special">+</span>
<span class="number">1</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="special">(</span><span class="identifier">a</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span>
<span class="special">(</span><span class="identifier">a</span>
<span class="special">+</span> <span class="identifier">b</span>
<span class="special">-</span> <span class="number">2</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="number">2</span> <span class="special">(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span>
<span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">1</span><span class="special">)/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">2</span><span class="special">)</span>
<span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span>
<span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis excess
</p>
</td>
<td>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/beta_dist_kurtosis.png"></span>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
<span class="number">3</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
parameter estimation
</p>
</td>
<td>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
alpha
</p>
<p>
from mean and variance
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
<span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
<span class="special">(</span><span class="number">1</span>
<span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span>
<span class="identifier">variance</span><span class="special">)-</span>
<span class="number">1</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
beta
</p>
<p>
from mean and variance
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="special">(</span><span class="number">1</span>
<span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span>
<span class="special">(((</span><span class="identifier">mean</span>
<span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span>
<span class="identifier">mean</span><span class="special">))</span>
<span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
The member functions <code class="computeroutput"><span class="identifier">find_alpha</span></code>
and <code class="computeroutput"><span class="identifier">find_beta</span></code>
</p>
<p>
from cdf and probability x
</p>
<p>
and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
or <code class="computeroutput"><span class="identifier">beta</span></code>
</p>
</td>
<td>
<p>
Implemented in terms of the inverse incomplete beta functions
</p>
<p>
<a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
and <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
respectively.
</p>
</td>
</tr>
<tr>
<td>
<p>
<code class="computeroutput"><span class="identifier">find_alpha</span></code>
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">probability</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
<code class="computeroutput"><span class="identifier">find_beta</span></code>
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">probability</span><span class="special">)</span></code>
</p>
</td>
</tr>
</tbody>
</table></div>
<a name="math_toolkit.dist.dist_ref.dists.beta_dist.references"></a><h5>
<a name="id1020480"></a>
<a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.references">References</a>
</h5>
<p>
<a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia
Beta distribution</a>
</p>
<p>
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm" target="_top">NIST
Exploratory Data Analysis</a>
</p>
<p>
<a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
MathWorld</a>
</p>
</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 John Maddock, Paul A. Bristow,
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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="bernoulli_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="binomial_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
