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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.beta_dist"></a><a class="link" href="beta_dist.html" title="Beta Distribution">Beta Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
<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 14. 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">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="../../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.svg" align="middle"></span>
</p>
<p>
If α = β = 1, then it is a __space <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 __space and β __space 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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.member_functions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.member_functions">Member
Functions</a>
</h5>
<h6>
<a name="math_toolkit.dist_ref.dists.beta_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.constructor"></a></span><a class="link" href="beta_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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>
<h6>
<a name="math_toolkit.dist_ref.dists.beta_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.parameter_accessors"></a></span><a class="link" href="beta_dist.html#math_toolkit.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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.parameter_estimators"></a></span><a class="link" href="beta_dist.html#math_toolkit.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 __space or β __space from presumed-known mean and variance.
</p>
<p>
The other pair estimates either α __space or β __space from the cdf and x.
</p>
<p>
It is also possible to estimate α __space and β __space 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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions"></a></span><a class="link" href="beta_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The 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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h5"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.applications"></a></span><a class="link" href="beta_dist.html#math_toolkit.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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h6"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.related_distributions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.related_distributions">Related
distributions</a>
</h5>
<p>
The beta distribution with both α __space 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 __space and β __space are integers.
</p>
<p>
With integer values of α __space and β __space 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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h7"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.accuracy"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.accuracy">Accuracy</a>
</h5>
<p>
This distribution is implemented using the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta
functions</a> <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>
and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete beta
functions</a> <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>;
please refer to these functions for information on accuracy.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h8"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.implementation"></a></span><a class="link" href="beta_dist.html#math_toolkit.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="../../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="../../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="../../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="../../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="../../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.svg"></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>
</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="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
and <a class="link" href="../../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>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h9"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.references"></a></span><a class="link" href="beta_dist.html#math_toolkit.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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></td>
</tr></table>
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