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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;15.&#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;15.&#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 <p>
 105           f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)
 106         </p>
 107 <p>
 108           where B(&#945;, &#946;) 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 &#945; = &#946; = 2 (blue line) is dome-shaped, and
 116           might be approximated by a symmetrical triangular distribution.
 117         </p>
 118 <p>
 119           <span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.svg" align="middle"></span>
 120         </p>
 121 <p>
 122           If &#945; = &#946; = 1, then it is a __space <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
 123           distribution</a>, equal to unity in the entire interval x = 0 to 1.
 124           If &#945; __space and &#946; __space are &lt; 1, then the pdf is U-shaped. If &#945; != &#946;, then
 125           the shape is asymmetric and could be approximated by a triangle whose apex
 126           is away from the centre (where x = half).
 127         </p>
 128 <h5>
 129 <a name="math_toolkit.dist_ref.dists.beta_dist.h0"></a>
 130           <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
 131           Functions</a>
 132         </h5>
 133 <h6>
 134 <a name="math_toolkit.dist_ref.dists.beta_dist.h1"></a>
 135           <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>
 136         </h6>
 137 <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>
 138 </pre>
 139 <p>
 140           Constructs a beta distribution with shape parameters <span class="emphasis"><em>alpha</em></span>
 141           and <span class="emphasis"><em>beta</em></span>.
 142         </p>
 143 <p>
 144           Requires alpha,beta &gt; 0,otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 145           is called. Note that technically the beta distribution is defined for alpha,beta
 146           &gt;= 0, but it's not clear whether any program can actually make use of
 147           that latitude or how many of the non-member functions can be usefully defined
 148           in that case. Therefore for now, we regard it as an error if alpha or beta
 149           is zero.
 150         </p>
 151 <p>
 152           For example:
 153         </p>
 154 <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>
 155 </pre>
 156 <p>
 157           Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow
 158           in the graph above).
 159         </p>
 160 <h6>
 161 <a name="math_toolkit.dist_ref.dists.beta_dist.h2"></a>
 162           <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
 163           Accessors</a>
 164         </h6>
 165 <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>
 166 </pre>
 167 <p>
 168           Returns the parameter <span class="emphasis"><em>alpha</em></span> from which this distribution
 169           was constructed.
 170         </p>
 171 <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>
 172 </pre>
 173 <p>
 174           Returns the parameter <span class="emphasis"><em>beta</em></span> from which this distribution
 175           was constructed.
 176         </p>
 177 <p>
 178           So for example:
 179         </p>
 180 <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>
 181 <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>
 182 <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>
 183 </pre>
 184 <h5>
 185 <a name="math_toolkit.dist_ref.dists.beta_dist.h3"></a>
 186           <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
 187           Estimators</a>
 188         </h5>
 189 <p>
 190           Two pairs of parameter estimators are provided.
 191         </p>
 192 <p>
 193           One estimates either &#945; __space or &#946; __space from presumed-known mean and variance.
 194         </p>
 195 <p>
 196           The other pair estimates either &#945; __space or &#946; __space from the cdf and x.
 197         </p>
 198 <p>
 199           It is also possible to estimate &#945; __space and &#946; __space from 'known' mode &amp;
 200           quantile. For example, calculators are provided by the <a href="http://www.ausvet.com.au/pprev/content.php?page=PPscript" target="_top">Pooled
 201           Prevalence Calculator</a> and <a href="http://www.epi.ucdavis.edu/diagnostictests/betabuster.html" target="_top">Beta
 202           Buster</a> but this is not yet implemented here.
 203         </p>
 204 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
 205   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
 206   <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
 207 </pre>
 208 <p>
 209           Returns the unique value of &#945; &#160; that corresponds to a beta distribution with
 210           mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
 211         </p>
 212 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
 213   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
 214   <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
 215 </pre>
 216 <p>
 217           Returns the unique value of &#946; &#160; that corresponds to a beta distribution with
 218           mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
 219         </p>
 220 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
 221   <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
 222   <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">//  x.</span>
 223   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf</span>
 224 </pre>
 225 <p>
 226           Returns the value of &#945; &#160; 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>.
 227         </p>
 228 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
 229   <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
 230   <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
 231   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
 232 </pre>
 233 <p>
 234           Returns the value of &#946; &#160; 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>.
 235         </p>
 236 <h5>
 237 <a name="math_toolkit.dist_ref.dists.beta_dist.h4"></a>
 238           <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
 239           Accessor Functions</a>
 240         </h5>
 241 <p>
 242           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 243           functions</a> that are generic to all distributions are supported:
 244           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 245           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 246           <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>,
 247           <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>,
 248           <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>,
 249           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 250           <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>,
 251           <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>.
 252         </p>
 253 <p>
 254           The formulae for calculating these are shown in the table below, and at
 255           <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 256           Mathworld</a>.
 257         </p>
 258 <h5>
 259 <a name="math_toolkit.dist_ref.dists.beta_dist.h5"></a>
 260           <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>
 261         </h5>
 262 <p>
 263           The beta distribution can be used to model events constrained to take place
 264           within an interval defined by a minimum and maximum value: so it is used
 265           in project management systems.
 266         </p>
 267 <p>
 268           It is also widely used in <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian
 269           statistical inference</a>.
 270         </p>
 271 <h5>
 272 <a name="math_toolkit.dist_ref.dists.beta_dist.h6"></a>
 273           <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
 274           distributions</a>
 275         </h5>
 276 <p>
 277           The beta distribution with both &#945;  __space and &#946; = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
 278           distribution</a>.
 279         </p>
 280 <p>
 281           The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular</a>
 282           is used when less precise information is available.
 283         </p>
 284 <p>
 285           The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
 286           distribution</a> is closely related when &#945;  __space and &#946;  __space are integers.
 287         </p>
 288 <p>
 289           With integer values of &#945;  __space and &#946;  __space the distribution B(i, j) is
 290           that of the j-th highest of a sample of i + j + 1 independent random variables
 291           uniformly distributed between 0 and 1. The cumulative probability from
 292           0 to x is thus the probability that the j-th highest value is less than
 293           x. Or it is the probability that at least i of the random variables are
 294           less than x, a probability given by summing over the <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
 295           Distribution</a> with its p parameter set to x.
 296         </p>
 297 <h5>
 298 <a name="math_toolkit.dist_ref.dists.beta_dist.h7"></a>
 299           <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>
 300         </h5>
 301 <p>
 302           This distribution is implemented using the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta
 303           functions</a> <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>
 304           and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete beta
 305           functions</a> <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
 306           and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>;
 307           please refer to these functions for information on accuracy.
 308         </p>
 309 <h5>
 310 <a name="math_toolkit.dist_ref.dists.beta_dist.h8"></a>
 311           <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>
 312         </h5>
 313 <p>
 314           In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 315           are the parameters &#945; &#160; and &#946;, <span class="emphasis"><em>x</em></span> is the random variable,
 316           <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 317         </p>
 318 <div class="informaltable"><table class="table">
 319 <colgroup>
 320 <col>
 321 <col>
 322 </colgroup>
 323 <thead><tr>
 324 <th>
 325                   <p>
 326                     Function
 327                   </p>
 328                 </th>
 329 <th>
 330                   <p>
 331                     Implementation Notes
 332                   </p>
 333                 </th>
 334 </tr></thead>
 335 <tbody>
 336 <tr>
 337 <td>
 338                   <p>
 339                     pdf
 340                   </p>
 341                 </td>
 342 <td>
 343                   <p>
 344                     f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)
 345                   </p>
 346                   <p>
 347                     Implemented using <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(a,
 348                     b, x).
 349                   </p>
 350                 </td>
 351 </tr>
 352 <tr>
 353 <td>
 354                   <p>
 355                     cdf
 356                   </p>
 357                 </td>
 358 <td>
 359                   <p>
 360                     Using the incomplete beta function <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(a,
 361                     b, x)
 362                   </p>
 363                 </td>
 364 </tr>
 365 <tr>
 366 <td>
 367                   <p>
 368                     cdf complement
 369                   </p>
 370                 </td>
 371 <td>
 372                   <p>
 373                     <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(a,
 374                     b, x)
 375                   </p>
 376                 </td>
 377 </tr>
 378 <tr>
 379 <td>
 380                   <p>
 381                     quantile
 382                   </p>
 383                 </td>
 384 <td>
 385                   <p>
 386                     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,
 387                     b, p)
 388                   </p>
 389                 </td>
 390 </tr>
 391 <tr>
 392 <td>
 393                   <p>
 394                     quantile from the complement
 395                   </p>
 396                 </td>
 397 <td>
 398                   <p>
 399                     <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(a,
 400                     b, q)
 401                   </p>
 402                 </td>
 403 </tr>
 404 <tr>
 405 <td>
 406                   <p>
 407                     mean
 408                   </p>
 409                 </td>
 410 <td>
 411                   <p>
 412                     <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>
 413                   </p>
 414                 </td>
 415 </tr>
 416 <tr>
 417 <td>
 418                   <p>
 419                     variance
 420                   </p>
 421                 </td>
 422 <td>
 423                   <p>
 424                     <code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
 425                     <span class="identifier">b</span> <span class="special">/</span>
 426                     <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>
 427                     <span class="identifier">b</span> <span class="special">+</span>
 428                     <span class="number">1</span><span class="special">)</span></code>
 429                   </p>
 430                 </td>
 431 </tr>
 432 <tr>
 433 <td>
 434                   <p>
 435                     mode
 436                   </p>
 437                 </td>
 438 <td>
 439                   <p>
 440                     <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>
 441                     <span class="special">(</span><span class="identifier">a</span>
 442                     <span class="special">+</span> <span class="identifier">b</span>
 443                     <span class="special">-</span> <span class="number">2</span><span class="special">)</span></code>
 444                   </p>
 445                 </td>
 446 </tr>
 447 <tr>
 448 <td>
 449                   <p>
 450                     skewness
 451                   </p>
 452                 </td>
 453 <td>
 454                   <p>
 455                     <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>
 456                     <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>
 457                     <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
 458                   </p>
 459                 </td>
 460 </tr>
 461 <tr>
 462 <td>
 463                   <p>
 464                     kurtosis excess
 465                   </p>
 466                 </td>
 467 <td>
 468                   <p>
 469                     <span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.svg"></span>
 470                   </p>
 471                 </td>
 472 </tr>
 473 <tr>
 474 <td>
 475                   <p>
 476                     kurtosis
 477                   </p>
 478                 </td>
 479 <td>
 480                   <p>
 481                     <code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
 482                     <span class="number">3</span></code>
 483                   </p>
 484                 </td>
 485 </tr>
 486 <tr>
 487 <td>
 488                   <p>
 489                     parameter estimation
 490                   </p>
 491                 </td>
 492 <td>
 493                 </td>
 494 </tr>
 495 <tr>
 496 <td>
 497                   <p>
 498                     alpha
 499                   </p>
 500                   <p>
 501                     from mean and variance
 502                   </p>
 503                 </td>
 504 <td>
 505                   <p>
 506                     <code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
 507                     <span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
 508                     <span class="special">(</span><span class="number">1</span>
 509                     <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span>
 510                     <span class="identifier">variance</span><span class="special">)-</span>
 511                     <span class="number">1</span><span class="special">)</span></code>
 512                   </p>
 513                 </td>
 514 </tr>
 515 <tr>
 516 <td>
 517                   <p>
 518                     beta
 519                   </p>
 520                   <p>
 521                     from mean and variance
 522                   </p>
 523                 </td>
 524 <td>
 525                   <p>
 526                     <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
 527                     <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span>
 528                     <span class="special">(((</span><span class="identifier">mean</span>
 529                     <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>
 530                     <span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
 531                   </p>
 532                 </td>
 533 </tr>
 534 <tr>
 535 <td>
 536                   <p>
 537                     The member functions <code class="computeroutput"><span class="identifier">find_alpha</span></code>
 538                     and <code class="computeroutput"><span class="identifier">find_beta</span></code>
 539                   </p>
 540                   <p>
 541                     from cdf and probability x
 542                   </p>
 543                   <p>
 544                     and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
 545                     or <code class="computeroutput"><span class="identifier">beta</span></code>
 546                   </p>
 547                 </td>
 548 <td>
 549                   <p>
 550                     Implemented in terms of the inverse incomplete beta functions
 551                   </p>
 552                   <p>
 553                     <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
 554                     and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
 555                     respectively.
 556                   </p>
 557                 </td>
 558 </tr>
 559 <tr>
 560 <td>
 561                   <p>
 562                     <code class="computeroutput"><span class="identifier">find_alpha</span></code>
 563                   </p>
 564                 </td>
 565 <td>
 566                   <p>
 567                     <code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span>
 568                     <span class="identifier">x</span><span class="special">,</span>
 569                     <span class="identifier">probability</span><span class="special">)</span></code>
 570                   </p>
 571                 </td>
 572 </tr>
 573 <tr>
 574 <td>
 575                   <p>
 576                     <code class="computeroutput"><span class="identifier">find_beta</span></code>
 577                   </p>
 578                 </td>
 579 <td>
 580                   <p>
 581                     <code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span>
 582                     <span class="identifier">x</span><span class="special">,</span>
 583                     <span class="identifier">probability</span><span class="special">)</span></code>
 584                   </p>
 585                 </td>
 586 </tr>
 587 </tbody>
 588 </table></div>
 589 <h5>
 590 <a name="math_toolkit.dist_ref.dists.beta_dist.h9"></a>
 591           <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>
 592         </h5>
 593 <p>
 594           <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia Beta
 595           distribution</a>
 596         </p>
 597 <p>
 598           <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm" target="_top">NIST
 599           Exploratory Data Analysis</a>
 600         </p>
 601 <p>
 602           <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 603           MathWorld</a>
 604         </p>
 605 </div>
 606 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 607 <td align="left"></td>
 608 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 609       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 610       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 611       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 612         Distributed under the Boost Software License, Version 1.0. (See accompanying
 613         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>)
 614       </p>
 615 </div></td>
 616 </tr></table>
 617 <hr>
 618 <div class="spirit-nav">
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