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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.beta_dist"></a><a class="link" href="beta_dist.html" title="Beta Distribution">Beta Distribution</a>
  28 </h4></div></div></div>
  29 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span></pre>
  30 <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>
  31
  32  <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  33            <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;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&lt;&gt;</a> <span class="special">&gt;</span>
  34 <span class="keyword">class</span> <span class="identifier">beta_distribution</span><span class="special">;</span>
  35
  36 <span class="comment">// typedef beta_distribution&lt;double&gt; beta;</span>
  37 <span class="comment">// Note that this is deliberately NOT provided,</span>
  38 <span class="comment">// to avoid a clash with the function name beta.</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <span class="keyword">class</span> <span class="identifier">beta_distribution</span>
  42 <span class="special">{</span>
  43 <span class="keyword">public</span><span class="special">:</span>
  44    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  45    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  46    <span class="comment">// Constructor from two shape parameters, alpha &amp; beta:</span>
  47    <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>
  48
  49    <span class="comment">// Parameter accessors:</span>
  50    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  51    <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  52
  53    <span class="comment">// Parameter estimators of alpha or beta from mean and variance.</span>
  54    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
  55      <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
  56      <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
  57
  58    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
  59      <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
  60      <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
  61
  62    <span class="comment">// Parameter estimators from</span>
  63    <span class="comment">// either alpha or beta, and x and probability.</span>
  64
  65    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
  66      <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
  67      <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">//  x.</span>
  68      <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// cdf</span>
  69
  70    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
  71      <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
  72      <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
  73      <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
  74 <span class="special">};</span>
  75
  76 <span class="special">}}</span> <span class="comment">// namespaces</span>
  77 </pre>
  78 <p>
  79           The class type <code class="computeroutput"><span class="identifier">beta_distribution</span></code>
  80           represents a <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
  81           </a> <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
  82           distribution function</a>.
  83         </p>
  84 <p>
  85           The <a href="http://mathworld.wolfram.com/BetaDistribution.htm" target="_top">beta
  86           distribution </a> is used as a <a href="http://en.wikipedia.org/wiki/Prior_distribution" target="_top">prior
  87           distribution</a> for binomial proportions in <a href="http://mathworld.wolfram.com/BayesianAnalysis.html" target="_top">Bayesian
  88           analysis</a>.
  89         </p>
  90 <p>
  91           See also: <a href="http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html" target="_top">beta
  92           distribution</a> and <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
  93           statistics</a>.
  94         </p>
  95 <p>
  96           How the beta distribution is used for <a href="http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt" target="_top">Bayesian
  97           analysis of one parameter models</a> is discussed by Jeff Grynaviski.
  98         </p>
  99 <p>
 100           The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
 101           density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
 102           distribution</a> defined on the interval [0,1] is given by:
 103         </p>
 104 <div class="blockquote"><blockquote class="blockquote"><p>
 105             <span class="serif_italic">f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)</span>
 106           </p></blockquote></div>
 107 <p>
 108           where <span class="serif_italic">B(&#945;, &#946;)</span> is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
 109           function</a>, implemented in this library as <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>.
 110           Division by the beta function ensures that the pdf is normalized to the
 111           range zero to unity.
 112         </p>
 113 <p>
 114           The following graph illustrates examples of the pdf for various values
 115           of the shape parameters. Note the <span class="emphasis"><em>&#945; = &#946; = 2</em></span> (blue line)
 116           is dome-shaped, and might be approximated by a symmetrical triangular distribution.
 117         </p>
 118 <div class="blockquote"><blockquote class="blockquote"><p>
 119             <span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.svg" align="middle"></span>
 120
 121           </p></blockquote></div>
 122 <p>
 123           If &#945; = &#946; = 1, then it is a &#8195;
 124 <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
 125           distribution</a>, equal to unity in the entire interval x = 0 to 1.
 126           If &#945; and &#946; are &lt; 1, then the pdf is U-shaped. If &#945; != &#946;, then the shape is
 127           asymmetric and could be approximated by a triangle whose apex is away from
 128           the centre (where x = half).
 129         </p>
 130 <h5>
 131 <a name="math_toolkit.dist_ref.dists.beta_dist.h0"></a>
 132           <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
 133           Functions</a>
 134         </h5>
 135 <h6>
 136 <a name="math_toolkit.dist_ref.dists.beta_dist.h1"></a>
 137           <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>
 138         </h6>
 139 <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>
 140 </pre>
 141 <p>
 142           Constructs a beta distribution with shape parameters <span class="emphasis"><em>alpha</em></span>
 143           and <span class="emphasis"><em>beta</em></span>.
 144         </p>
 145 <p>
 146           Requires alpha,beta &gt; 0,otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 147           is called. Note that technically the beta distribution is defined for alpha,beta
 148           &gt;= 0, but it's not clear whether any program can actually make use of
 149           that latitude or how many of the non-member functions can be usefully defined
 150           in that case. Therefore for now, we regard it as an error if alpha or beta
 151           is zero.
 152         </p>
 153 <p>
 154           For example:
 155         </p>
 156 <pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special">&lt;&gt;</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>
 157 </pre>
 158 <p>
 159           Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow
 160           in the graph above).
 161         </p>
 162 <h6>
 163 <a name="math_toolkit.dist_ref.dists.beta_dist.h2"></a>
 164           <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
 165           Accessors</a>
 166         </h6>
 167 <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>
 168 </pre>
 169 <p>
 170           Returns the parameter <span class="emphasis"><em>alpha</em></span> from which this distribution
 171           was constructed.
 172         </p>
 173 <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>
 174 </pre>
 175 <p>
 176           Returns the parameter <span class="emphasis"><em>beta</em></span> from which this distribution
 177           was constructed.
 178         </p>
 179 <p>
 180           So for example:
 181         </p>
 182 <pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special">&lt;&gt;</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>
 183 <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>
 184 <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>
 185 </pre>
 186 <h5>
 187 <a name="math_toolkit.dist_ref.dists.beta_dist.h3"></a>
 188           <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
 189           Estimators</a>
 190         </h5>
 191 <p>
 192           Two pairs of parameter estimators are provided.
 193         </p>
 194 <p>
 195           One estimates either &#945;  or &#946;
 196 from presumed-known mean and variance.
 197         </p>
 198 <p>
 199           The other pair estimates either &#945; or &#946; from the cdf and x.
 200         </p>
 201 <p>
 202           It is also possible to estimate &#945; and &#946;  from 'known' mode &amp; quantile. For
 203           example, calculators are provided by the <a href="http://www.ausvet.com.au/pprev/content.php?page=PPscript" target="_top">Pooled
 204           Prevalence Calculator</a> and <a href="http://www.epi.ucdavis.edu/diagnostictests/betabuster.html" target="_top">Beta
 205           Buster</a> but this is not yet implemented here.
 206         </p>
 207 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
 208   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
 209   <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
 210 </pre>
 211 <p>
 212           Returns the unique value of &#945; that corresponds to a beta distribution with
 213           mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
 214         </p>
 215 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
 216   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
 217   <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
 218 </pre>
 219 <p>
 220           Returns the unique value of &#946; that corresponds to a beta distribution with
 221           mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
 222         </p>
 223 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
 224   <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
 225   <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">//  x.</span>
 226   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf</span>
 227 </pre>
 228 <p>
 229           Returns the value of &#945; that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</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>.
 230         </p>
 231 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
 232   <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
 233   <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
 234   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
 235 </pre>
 236 <p>
 237           Returns the value of &#946; that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</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>.
 238         </p>
 239 <h5>
 240 <a name="math_toolkit.dist_ref.dists.beta_dist.h4"></a>
 241           <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
 242           Accessor Functions</a>
 243         </h5>
 244 <p>
 245           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 246           functions</a> that are generic to all distributions are supported:
 247           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 248           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 249           <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>,
 250           <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>,
 251           <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>,
 252           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 253           <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>,
 254           <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>.
 255         </p>
 256 <p>
 257           The formulae for calculating these are shown in the table below, and at
 258           <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 259           Mathworld</a>.
 260         </p>
 261 <h5>
 262 <a name="math_toolkit.dist_ref.dists.beta_dist.h5"></a>
 263           <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>
 264         </h5>
 265 <p>
 266           The beta distribution can be used to model events constrained to take place
 267           within an interval defined by a minimum and maximum value: so it is used
 268           in project management systems.
 269         </p>
 270 <p>
 271           It is also widely used in <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian
 272           statistical inference</a>.
 273         </p>
 274 <h5>
 275 <a name="math_toolkit.dist_ref.dists.beta_dist.h6"></a>
 276           <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
 277           distributions</a>
 278         </h5>
 279 <p>
 280           The beta distribution with both &#945; and &#946; = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
 281           distribution</a>.
 282         </p>
 283 <p>
 284           The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular</a>
 285           is used when less precise information is available.
 286         </p>
 287 <p>
 288           The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
 289           distribution</a> is closely related when &#945; and &#946;  are integers.
 290         </p>
 291 <p>
 292           With integer values of &#945;  and &#946; the distribution B(i, j) is that of the j-th
 293           highest of a sample of i + j + 1 independent random variables uniformly
 294           distributed between 0 and 1. The cumulative probability from 0 to x is
 295           thus the probability that the j-th highest value is less than x. Or it
 296           is the probability that at least i of the random variables are less than
 297           x, a probability given by summing over the <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
 298           Distribution</a> with its p parameter set to x.
 299         </p>
 300 <h5>
 301 <a name="math_toolkit.dist_ref.dists.beta_dist.h7"></a>
 302           <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>
 303         </h5>
 304 <p>
 305           This distribution is implemented using the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta
 306           functions</a> <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>
 307           and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete beta
 308           functions</a> <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
 309           and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>;
 310           please refer to these functions for information on accuracy.
 311         </p>
 312 <h5>
 313 <a name="math_toolkit.dist_ref.dists.beta_dist.h8"></a>
 314           <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>
 315         </h5>
 316 <p>
 317           In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 318           are the parameters &#945; and &#946;, <span class="emphasis"><em>x</em></span> is the random variable,
 319           <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 320         </p>
 321 <div class="informaltable"><table class="table">
 322 <colgroup>
 323 <col>
 324 <col>
 325 </colgroup>
 326 <thead><tr>
 327 <th>
 328                   <p>
 329                     Function
 330                   </p>
 331                 </th>
 332 <th>
 333                   <p>
 334                     Implementation Notes
 335                   </p>
 336                 </th>
 337 </tr></thead>
 338 <tbody>
 339 <tr>
 340 <td>
 341                   <p>
 342                     pdf
 343                   </p>
 344                 </td>
 345 <td>
 346                   <p>
 347                     <span class="serif_italic">f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)</span>
 348                   </p>
 349                   <p>
 350                     Implemented using <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(a,
 351                     b, x).
 352                   </p>
 353                 </td>
 354 </tr>
 355 <tr>
 356 <td>
 357                   <p>
 358                     cdf
 359                   </p>
 360                 </td>
 361 <td>
 362                   <p>
 363                     Using the incomplete beta function <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(a,
 364                     b, x)
 365                   </p>
 366                 </td>
 367 </tr>
 368 <tr>
 369 <td>
 370                   <p>
 371                     cdf complement
 372                   </p>
 373                 </td>
 374 <td>
 375                   <p>
 376                     <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(a,
 377                     b, x)
 378                   </p>
 379                 </td>
 380 </tr>
 381 <tr>
 382 <td>
 383                   <p>
 384                     quantile
 385                   </p>
 386                 </td>
 387 <td>
 388                   <p>
 389                     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,
 390                     b, p)
 391                   </p>
 392                 </td>
 393 </tr>
 394 <tr>
 395 <td>
 396                   <p>
 397                     quantile from the complement
 398                   </p>
 399                 </td>
 400 <td>
 401                   <p>
 402                     <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(a,
 403                     b, q)
 404                   </p>
 405                 </td>
 406 </tr>
 407 <tr>
 408 <td>
 409                   <p>
 410                     mean
 411                   </p>
 412                 </td>
 413 <td>
 414                   <p>
 415                     <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>
 416                   </p>
 417                 </td>
 418 </tr>
 419 <tr>
 420 <td>
 421                   <p>
 422                     variance
 423                   </p>
 424                 </td>
 425 <td>
 426                   <p>
 427                     <code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
 428                     <span class="identifier">b</span> <span class="special">/</span>
 429                     <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>
 430                     <span class="identifier">b</span> <span class="special">+</span>
 431                     <span class="number">1</span><span class="special">)</span></code>
 432                   </p>
 433                 </td>
 434 </tr>
 435 <tr>
 436 <td>
 437                   <p>
 438                     mode
 439                   </p>
 440                 </td>
 441 <td>
 442                   <p>
 443                     <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>
 444                     <span class="special">(</span><span class="identifier">a</span>
 445                     <span class="special">+</span> <span class="identifier">b</span>
 446                     <span class="special">-</span> <span class="number">2</span><span class="special">)</span></code>
 447                   </p>
 448                 </td>
 449 </tr>
 450 <tr>
 451 <td>
 452                   <p>
 453                     skewness
 454                   </p>
 455                 </td>
 456 <td>
 457                   <p>
 458                     <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>
 459                     <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>
 460                     <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
 461                   </p>
 462                 </td>
 463 </tr>
 464 <tr>
 465 <td>
 466                   <p>
 467                     kurtosis excess
 468                   </p>
 469                 </td>
 470 <td>
 471                   <div class="blockquote"><blockquote class="blockquote"><p>
 472                       <span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.svg"></span>
 473
 474                     </p></blockquote></div>
 475                 </td>
 476 </tr>
 477 <tr>
 478 <td>
 479                   <p>
 480                     kurtosis
 481                   </p>
 482                 </td>
 483 <td>
 484                   <p>
 485                     <code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
 486                     <span class="number">3</span></code>
 487                   </p>
 488                 </td>
 489 </tr>
 490 <tr>
 491 <td>
 492                   <p>
 493                     parameter estimation
 494                   </p>
 495                 </td>
 496 <td>
 497                 </td>
 498 </tr>
 499 <tr>
 500 <td>
 501                   <p>
 502                     alpha (from mean and variance)
 503                   </p>
 504                 </td>
 505 <td>
 506                   <p>
 507                     <code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
 508                     <span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
 509                     <span class="special">(</span><span class="number">1</span>
 510                     <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span>
 511                     <span class="identifier">variance</span><span class="special">)-</span>
 512                     <span class="number">1</span><span class="special">)</span></code>
 513                   </p>
 514                 </td>
 515 </tr>
 516 <tr>
 517 <td>
 518                   <p>
 519                     beta (from mean and variance)
 520                   </p>
 521                 </td>
 522 <td>
 523                   <p>
 524                     <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
 525                     <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span>
 526                     <span class="special">(((</span><span class="identifier">mean</span>
 527                     <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>
 528                     <span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
 529                   </p>
 530                 </td>
 531 </tr>
 532 <tr>
 533 <td>
 534                   <p>
 535                     The member functions <code class="computeroutput"><span class="identifier">find_alpha</span></code>
 536                     and <code class="computeroutput"><span class="identifier">find_beta</span></code>
 537                   </p>
 538                   <p>
 539                     from cdf and probability x
 540                   </p>
 541                   <p>
 542                     and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
 543                     or <code class="computeroutput"><span class="identifier">beta</span></code>
 544                   </p>
 545                 </td>
 546 <td>
 547                   <p>
 548                     Implemented in terms of the inverse incomplete beta functions
 549                   </p>
 550                   <p>
 551                     <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
 552                     and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
 553                     respectively.
 554                   </p>
 555                 </td>
 556 </tr>
 557 <tr>
 558 <td>
 559                   <p>
 560                     <code class="computeroutput"><span class="identifier">find_alpha</span></code>
 561                   </p>
 562                 </td>
 563 <td>
 564                   <p>
 565                     <code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span>
 566                     <span class="identifier">x</span><span class="special">,</span>
 567                     <span class="identifier">probability</span><span class="special">)</span></code>
 568                   </p>
 569                 </td>
 570 </tr>
 571 <tr>
 572 <td>
 573                   <p>
 574                     <code class="computeroutput"><span class="identifier">find_beta</span></code>
 575                   </p>
 576                 </td>
 577 <td>
 578                   <p>
 579                     <code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span>
 580                     <span class="identifier">x</span><span class="special">,</span>
 581                     <span class="identifier">probability</span><span class="special">)</span></code>
 582                   </p>
 583                 </td>
 584 </tr>
 585 </tbody>
 586 </table></div>
 587 <h5>
 588 <a name="math_toolkit.dist_ref.dists.beta_dist.h9"></a>
 589           <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>
 590         </h5>
 591 <p>
 592           <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia Beta
 593           distribution</a>
 594         </p>
 595 <p>
 596           <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm" target="_top">NIST
 597           Exploratory Data Analysis</a>
 598         </p>
 599 <p>
 600           <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 601           MathWorld</a>
 602         </p>
 603 </div>
 604 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 605 <td align="left"></td>
 606 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 607       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 608       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 609       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 610       Daryle Walker and Xiaogang Zhang<p>
 611         Distributed under the Boost Software License, Version 1.0. (See accompanying
 612         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>)
 613       </p>
 614 </div></td>
 615 </tr></table>
 616 <hr>
 617 <div class="spirit-nav">
 618 <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>
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