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  25 <div class="section math_toolkit_dist_dist_ref_dists_beta_dist">
  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.dist.dist_ref.dists.beta_dist"></a><a class="link" href="beta_dist.html" title="Beta Distribution">Beta
  28           Distribution</a>
  29 </h5></div></div></div>
  30 <p>
  31 </p>
  32 <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>
  33 <p>
  34           </p>
  35 <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>
  36
  37  <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>
  38            <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a>   <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">beta_distribution</span><span class="special">;</span>
  40
  41 <span class="comment">// typedef beta_distribution&lt;double&gt; beta;</span>
  42 <span class="comment">// Note that this is deliberately NOT provided,</span>
  43 <span class="comment">// to avoid a clash with the function name beta.</span>
  44
  45 <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="Policies">Policy</a><span class="special">&gt;</span>
  46 <span class="keyword">class</span> <span class="identifier">beta_distribution</span>
  47 <span class="special">{</span>
  48 <span class="keyword">public</span><span class="special">:</span>
  49    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  50    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  51    <span class="comment">// Constructor from two shape parameters, alpha &amp; beta:</span>
  52    <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>
  53
  54    <span class="comment">// Parameter accessors:</span>
  55    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  56    <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  57
  58    <span class="comment">// Parameter estimators of alpha or beta from mean and variance.</span>
  59    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
  60      <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
  61      <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
  62
  63    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
  64      <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
  65      <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
  66
  67    <span class="comment">// Parameter estimators from from</span>
  68    <span class="comment">// either alpha or beta, and x and probability.</span>
  69
  70    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
  71      <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
  72      <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">//  x.</span>
  73      <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// cdf</span>
  74
  75    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
  76      <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
  77      <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
  78      <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
  79 <span class="special">};</span>
  80
  81 <span class="special">}}</span> <span class="comment">// namespaces</span>
  82 </pre>
  83 <p>
  84             The class type <code class="computeroutput"><span class="identifier">beta_distribution</span></code>
  85             represents a <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
  86             </a> <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
  87             distribution function</a>.
  88           </p>
  89 <p>
  90             The <a href="http://mathworld.wolfram.com/BetaDistribution.htm" target="_top">beta
  91             distribution </a> is used as a <a href="http://en.wikipedia.org/wiki/Prior_distribution" target="_top">prior
  92             distribution</a> for binomial proportions in <a href="http://mathworld.wolfram.com/BayesianAnalysis.html" target="_top">Bayesian
  93             analysis</a>.
  94           </p>
  95 <p>
  96             See also: <a href="http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html" target="_top">beta
  97             distribution</a> and <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
  98             statistics</a>.
  99           </p>
 100 <p>
 101             How the beta distribution is used for <a href="http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt" target="_top">Bayesian
 102             analysis of one parameter models</a> is discussed by Jeff Grynaviski.
 103           </p>
 104 <p>
 105             The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
 106             density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
 107             distribution</a> defined on the interval [0,1] is given by:
 108           </p>
 109 <p>
 110             f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)
 111           </p>
 112 <p>
 113             where B(&#945;, &#946;) is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
 114             function</a>, implemented in this library as <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta</a>.
 115             Division by the beta function ensures that the pdf is normalized to the
 116             range zero to unity.
 117           </p>
 118 <p>
 119             The following graph illustrates examples of the pdf for various values
 120             of the shape parameters. Note the &#945; = &#946; = 2 (blue line) is dome-shaped, and
 121             might be approximated by a symmetrical triangular distribution.
 122           </p>
 123 <p>
 124             <span class="inlinemediaobject"><img src="../../../../../graphs/beta_pdf.png" align="middle"></span>
 125           </p>
 126 <p>
 127             If &#945; = &#946; = 1, then it is a __space <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
 128             distribution</a>, equal to unity in the entire interval x = 0 to
 129             1. If &#945; __space and &#946; __space are &lt; 1, then the pdf is U-shaped. If &#945; !=
 130             &#946;, then the shape is asymmetric and could be approximated by a triangle
 131             whose apex is away from the centre (where x = half).
 132           </p>
 133 <h5>
 134 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h0"></a>
 135             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.member_functions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.member_functions">Member
 136             Functions</a>
 137           </h5>
 138 <h6>
 139 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h1"></a>
 140             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.constructor"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.constructor">Constructor</a>
 141           </h6>
 142 <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>
 143 </pre>
 144 <p>
 145             Constructs a beta distribution with shape parameters <span class="emphasis"><em>alpha</em></span>
 146             and <span class="emphasis"><em>beta</em></span>.
 147           </p>
 148 <p>
 149             Requires alpha,beta &gt; 0,otherwise <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
 150             is called. Note that technically the beta distribution is defined for
 151             alpha,beta &gt;= 0, but it's not clear whether any program can actually
 152             make use of that latitude or how many of the non-member functions can
 153             be usefully defined in that case. Therefore for now, we regard it as
 154             an error if alpha or beta is zero.
 155           </p>
 156 <p>
 157             For example:
 158           </p>
 159 <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>
 160 </pre>
 161 <p>
 162             Constructs a the beta distribution with alpha=2 and beta=5 (shown in
 163             yellow in the graph above).
 164           </p>
 165 <h6>
 166 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h2"></a>
 167             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.parameter_accessors"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.parameter_accessors">Parameter
 168             Accessors</a>
 169           </h6>
 170 <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>
 171 </pre>
 172 <p>
 173             Returns the parameter <span class="emphasis"><em>alpha</em></span> from which this distribution
 174             was constructed.
 175           </p>
 176 <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>
 177 </pre>
 178 <p>
 179             Returns the parameter <span class="emphasis"><em>beta</em></span> from which this distribution
 180             was constructed.
 181           </p>
 182 <p>
 183             So for example:
 184           </p>
 185 <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>
 186 <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>
 187 <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>
 188 </pre>
 189 <h5>
 190 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h3"></a>
 191             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.parameter_estimators"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.parameter_estimators">Parameter
 192             Estimators</a>
 193           </h5>
 194 <p>
 195             Two pairs of parameter estimators are provided.
 196           </p>
 197 <p>
 198             One estimates either &#945; __space or &#946; __space from presumed-known mean and
 199             variance.
 200           </p>
 201 <p>
 202             The other pair estimates either &#945; __space or &#946; __space from the cdf and x.
 203           </p>
 204 <p>
 205             It is also possible to estimate &#945; __space and &#946; __space from 'known' mode
 206             &amp; quantile. For example, calculators are provided by the <a href="http://www.ausvet.com.au/pprev/content.php?page=PPscript" target="_top">Pooled
 207             Prevalence Calculator</a> and <a href="http://www.epi.ucdavis.edu/diagnostictests/betabuster.html" target="_top">Beta
 208             Buster</a> but this is not yet implemented here.
 209           </p>
 210 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
 211   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
 212   <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
 213 </pre>
 214 <p>
 215             Returns the unique value of &#945; &#160; that corresponds to a beta distribution with
 216             mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
 217           </p>
 218 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
 219   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
 220   <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
 221 </pre>
 222 <p>
 223             Returns the unique value of &#946; &#160; that corresponds to a beta distribution with
 224             mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
 225           </p>
 226 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
 227   <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
 228   <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">//  x.</span>
 229   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf</span>
 230 </pre>
 231 <p>
 232             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>.
 233           </p>
 234 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
 235   <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
 236   <span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
 237   <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
 238 </pre>
 239 <p>
 240             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>.
 241           </p>
 242 <h5>
 243 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h4"></a>
 244             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.non_member_accessor_functions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.non_member_accessor_functions">Non-member
 245             Accessor Functions</a>
 246           </h5>
 247 <p>
 248             All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
 249             accessor functions</a> that are generic to all distributions are supported:
 250             <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>,
 251             <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard
 252             Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>,
 253             <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>,
 254             <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>,
 255             <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>,
 256             <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>,
 257             <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>.
 258           </p>
 259 <p>
 260             The formulae for calculating these are shown in the table below, and
 261             at <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 262             Mathworld</a>.
 263           </p>
 264 <h5>
 265 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h5"></a>
 266             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.applications"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.applications">Applications</a>
 267           </h5>
 268 <p>
 269             The beta distribution can be used to model events constrained to take
 270             place within an interval defined by a minimum and maximum value: so it
 271             is used in project management systems.
 272           </p>
 273 <p>
 274             It is also widely used in <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian
 275             statistical inference</a>.
 276           </p>
 277 <h5>
 278 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h6"></a>
 279             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.related_distributions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.related_distributions">Related
 280             distributions</a>
 281           </h5>
 282 <p>
 283             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
 284             distribution</a>.
 285           </p>
 286 <p>
 287             The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular</a>
 288             is used when less precise information is available.
 289           </p>
 290 <p>
 291             The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
 292             distribution</a> is closely related when &#945;  __space and &#946;  __space are
 293             integers.
 294           </p>
 295 <p>
 296             With integer values of &#945;  __space and &#946;  __space the distribution B(i, j) is
 297             that of the j-th highest of a sample of i + j + 1 independent random
 298             variables uniformly distributed between 0 and 1. The cumulative probability
 299             from 0 to x is thus the probability that the j-th highest value is less
 300             than x. Or it is the probability that that at least i of the random variables
 301             are less than x, a probability given by summing over the <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
 302             Distribution</a> with its p parameter set to x.
 303           </p>
 304 <h5>
 305 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h7"></a>
 306             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.accuracy"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.accuracy">Accuracy</a>
 307           </h5>
 308 <p>
 309             This distribution is implemented using the <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta
 310             functions</a> <a class="link" href="../../../special/sf_beta/beta_function.html" title="Beta">beta</a>
 311             and <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
 312             beta functions</a> <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
 313             and <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>;
 314             please refer to these functions for information on accuracy.
 315           </p>
 316 <h5>
 317 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h8"></a>
 318             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.implementation"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.implementation">Implementation</a>
 319           </h5>
 320 <p>
 321             In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 322             are the parameters &#945; &#160; and &#946;, <span class="emphasis"><em>x</em></span> is the random variable,
 323             <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 324           </p>
 325 <div class="informaltable"><table class="table">
 326 <colgroup>
 327 <col>
 328 <col>
 329 </colgroup>
 330 <thead><tr>
 331 <th>
 332                     <p>
 333                       Function
 334                     </p>
 335                   </th>
 336 <th>
 337                     <p>
 338                       Implementation Notes
 339                     </p>
 340                   </th>
 341 </tr></thead>
 342 <tbody>
 343 <tr>
 344 <td>
 345                     <p>
 346                       pdf
 347                     </p>
 348                   </td>
 349 <td>
 350                     <p>
 351                       f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)
 352                     </p>
 353                     <p>
 354                       Implemented using <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(a,
 355                       b, x).
 356                     </p>
 357                   </td>
 358 </tr>
 359 <tr>
 360 <td>
 361                     <p>
 362                       cdf
 363                     </p>
 364                   </td>
 365 <td>
 366                     <p>
 367                       Using the incomplete beta function <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(a,
 368                       b, x)
 369                     </p>
 370                   </td>
 371 </tr>
 372 <tr>
 373 <td>
 374                     <p>
 375                       cdf complement
 376                     </p>
 377                   </td>
 378 <td>
 379                     <p>
 380                       <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(a,
 381                       b, x)
 382                     </p>
 383                   </td>
 384 </tr>
 385 <tr>
 386 <td>
 387                     <p>
 388                       quantile
 389                     </p>
 390                   </td>
 391 <td>
 392                     <p>
 393                       Using the inverse incomplete beta function <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(a,
 394                       b, p)
 395                     </p>
 396                   </td>
 397 </tr>
 398 <tr>
 399 <td>
 400                     <p>
 401                       quantile from the complement
 402                     </p>
 403                   </td>
 404 <td>
 405                     <p>
 406                       <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(a,
 407                       b, q)
 408                     </p>
 409                   </td>
 410 </tr>
 411 <tr>
 412 <td>
 413                     <p>
 414                       mean
 415                     </p>
 416                   </td>
 417 <td>
 418                     <p>
 419                       <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>
 420                     </p>
 421                   </td>
 422 </tr>
 423 <tr>
 424 <td>
 425                     <p>
 426                       variance
 427                     </p>
 428                   </td>
 429 <td>
 430                     <p>
 431                       <code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
 432                       <span class="identifier">b</span> <span class="special">/</span>
 433                       <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>
 434                       <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span>
 435                       <span class="identifier">b</span> <span class="special">+</span>
 436                       <span class="number">1</span><span class="special">)</span></code>
 437                     </p>
 438                   </td>
 439 </tr>
 440 <tr>
 441 <td>
 442                     <p>
 443                       mode
 444                     </p>
 445                   </td>
 446 <td>
 447                     <p>
 448                       <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>
 449                       <span class="special">(</span><span class="identifier">a</span>
 450                       <span class="special">+</span> <span class="identifier">b</span>
 451                       <span class="special">-</span> <span class="number">2</span><span class="special">)</span></code>
 452                     </p>
 453                   </td>
 454 </tr>
 455 <tr>
 456 <td>
 457                     <p>
 458                       skewness
 459                     </p>
 460                   </td>
 461 <td>
 462                     <p>
 463                       <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>
 464                       <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>
 465                       <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span>
 466                       <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
 467                     </p>
 468                   </td>
 469 </tr>
 470 <tr>
 471 <td>
 472                     <p>
 473                       kurtosis excess
 474                     </p>
 475                   </td>
 476 <td>
 477                     <p>
 478                       <span class="inlinemediaobject"><img src="../../../../../equations/beta_dist_kurtosis.png"></span>
 479                     </p>
 480                   </td>
 481 </tr>
 482 <tr>
 483 <td>
 484                     <p>
 485                       kurtosis
 486                     </p>
 487                   </td>
 488 <td>
 489                     <p>
 490                       <code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
 491                       <span class="number">3</span></code>
 492                     </p>
 493                   </td>
 494 </tr>
 495 <tr>
 496 <td>
 497                     <p>
 498                       parameter estimation
 499                     </p>
 500                   </td>
 501 <td>
 502                   </td>
 503 </tr>
 504 <tr>
 505 <td>
 506                     <p>
 507                       alpha
 508                     </p>
 509                     <p>
 510                       from mean and variance
 511                     </p>
 512                   </td>
 513 <td>
 514                     <p>
 515                       <code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
 516                       <span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
 517                       <span class="special">(</span><span class="number">1</span>
 518                       <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span>
 519                       <span class="identifier">variance</span><span class="special">)-</span>
 520                       <span class="number">1</span><span class="special">)</span></code>
 521                     </p>
 522                   </td>
 523 </tr>
 524 <tr>
 525 <td>
 526                     <p>
 527                       beta
 528                     </p>
 529                     <p>
 530                       from mean and variance
 531                     </p>
 532                   </td>
 533 <td>
 534                     <p>
 535                       <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
 536                       <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span>
 537                       <span class="special">(((</span><span class="identifier">mean</span>
 538                       <span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span>
 539                       <span class="identifier">mean</span><span class="special">))</span>
 540                       <span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
 541                     </p>
 542                   </td>
 543 </tr>
 544 <tr>
 545 <td>
 546                     <p>
 547                       The member functions <code class="computeroutput"><span class="identifier">find_alpha</span></code>
 548                       and <code class="computeroutput"><span class="identifier">find_beta</span></code>
 549                     </p>
 550                     <p>
 551                       from cdf and probability x
 552                     </p>
 553                     <p>
 554                       and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
 555                       or <code class="computeroutput"><span class="identifier">beta</span></code>
 556                     </p>
 557                   </td>
 558 <td>
 559                     <p>
 560                       Implemented in terms of the inverse incomplete beta functions
 561                     </p>
 562                     <p>
 563                       <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
 564                       and <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
 565                       respectively.
 566                     </p>
 567                   </td>
 568 </tr>
 569 <tr>
 570 <td>
 571                     <p>
 572                       <code class="computeroutput"><span class="identifier">find_alpha</span></code>
 573                     </p>
 574                   </td>
 575 <td>
 576                     <p>
 577                       <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>
 578                     </p>
 579                   </td>
 580 </tr>
 581 <tr>
 582 <td>
 583                     <p>
 584                       <code class="computeroutput"><span class="identifier">find_beta</span></code>
 585                     </p>
 586                   </td>
 587 <td>
 588                     <p>
 589                       <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>
 590                     </p>
 591                   </td>
 592 </tr>
 593 </tbody>
 594 </table></div>
 595 <h5>
 596 <a name="math_toolkit.dist.dist_ref.dists.beta_dist.h9"></a>
 597             <span><a name="math_toolkit.dist.dist_ref.dists.beta_dist.references"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist.dist_ref.dists.beta_dist.references">References</a>
 598           </h5>
 599 <p>
 600             <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia
 601             Beta distribution</a>
 602           </p>
 603 <p>
 604             <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm" target="_top">NIST
 605             Exploratory Data Analysis</a>
 606           </p>
 607 <p>
 608             <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 609             MathWorld</a>
 610           </p>
 611 </div>
 612 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 613 <td align="left"></td>
 614 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 615       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 616         Distributed under the Boost Software License, Version 1.0. (See accompanying
 617         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>)
 618       </p>
 619 </div></td>
 620 </tr></table>
 621 <hr>
 622 <div class="spirit-nav">
 623 <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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