1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Beta Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="bernoulli_dist.html" title="Bernoulli Distribution"> 10<link rel="next" href="binomial_dist.html" title="Binomial Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<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> 24</div> 25<div class="section"> 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"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> 34<span class="keyword">class</span> <span class="identifier">beta_distribution</span><span class="special">;</span> 35 36<span class="comment">// typedef beta_distribution<double> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 & 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;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)</span> 106 </p></blockquote></div> 107<p> 108 where <span class="serif_italic">B(α, β)</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>α = β = 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 α = β = 1, then it is a 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 α and β are < 1, then the pdf is U-shaped. If α != β, 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 > 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 >= 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"><></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"><></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 α or β 196from presumed-known mean and variance. 197 </p> 198<p> 199 The other pair estimates either α or β from the cdf and x. 200 </p> 201<p> 202 It is also possible to estimate α and β from 'known' mode & 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 α 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 β 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 α that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>. 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 β that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>. 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 α and β = 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 α and β are integers. 290 </p> 291<p> 292 With integer values of α and β 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 α and β, <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;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)</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 © 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 Råde, 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> 619</div> 620</body> 621</html> 622
