ceph/src/boost/libs/math/doc/html/math_toolkit/dist_ref/dists/bernoulli_dist.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Bernoulli Distribution</title>
   5 <link rel="stylesheet" href="../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
   7 <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.1">
   8 <link rel="up" href="../dists.html" title="Distributions">
   9 <link rel="prev" href="arcine_dist.html" title="Arcsine Distribution">
  10 <link rel="next" href="beta_dist.html" title="Beta 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="arcine_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="beta_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.bernoulli_dist"></a><a class="link" href="bernoulli_dist.html" title="Bernoulli Distribution">Bernoulli
  28         Distribution</a>
  29 </h4></div></div></div>
  30 <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">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <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>
  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">bernoulli_distribution</span><span class="special">;</span>
  35
  36  <span class="keyword">typedef</span> <span class="identifier">bernoulli_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">bernoulli</span><span class="special">;</span>
  37
  38  <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>
  39  <span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span>
  40  <span class="special">{</span>
  41  <span class="keyword">public</span><span class="special">:</span>
  42     <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  43     <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  44
  45     <span class="identifier">bernoulli_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> <span class="comment">// Constructor.</span>
  46     <span class="comment">// Accessor function.</span>
  47     <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span>
  48     <span class="comment">// Probability of success (as a fraction).</span>
  49  <span class="special">};</span>
  50 <span class="special">}}</span> <span class="comment">// namespaces</span>
  51 </pre>
  52 <p>
  53           The Bernoulli distribution is a discrete distribution of the outcome of
  54           a single trial with only two results, 0 (failure) or 1 (success), with
  55           a probability of success p.
  56         </p>
  57 <p>
  58           The Bernoulli distribution is the simplest building block on which other
  59           discrete distributions of sequences of independent Bernoulli trials can
  60           be based.
  61         </p>
  62 <p>
  63           The Bernoulli is the binomial distribution (k = 1, p) with only one trial.
  64         </p>
  65 <p>
  66           <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
  67           density function pdf</a> f(0) = 1 - p, f(1) = p. <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative
  68           distribution function</a> D(k) = if (k == 0) 1 - p else 1.
  69         </p>
  70 <p>
  71           The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
  72           density function pdf</a> varies with the outcome of the single trial:
  73         </p>
  74 <p>
  75           <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_pdf.svg" align="middle"></span>
  76         </p>
  77 <p>
  78           and the <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative
  79           distribution function</a>
  80         </p>
  81 <p>
  82           <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_cdf.svg" align="middle"></span>
  83         </p>
  84 <h5>
  85 <a name="math_toolkit.dist_ref.dists.bernoulli_dist.h0"></a>
  86           <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.member_functions"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.member_functions">Member
  87           Functions</a>
  88         </h5>
  89 <pre class="programlisting"><span class="identifier">bernoulli_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
  90 </pre>
  91 <p>
  92           Constructs a <a href="http://en.wikipedia.org/wiki/bernoulli_distribution" target="_top">bernoulli
  93           distribution</a> with success_fraction <span class="emphasis"><em>p</em></span>.
  94         </p>
  95 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span>
  96 </pre>
  97 <p>
  98           Returns the <span class="emphasis"><em>success_fraction</em></span> parameter of this distribution.
  99         </p>
 100 <h5>
 101 <a name="math_toolkit.dist_ref.dists.bernoulli_dist.h1"></a>
 102           <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.non_member_accessors"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.non_member_accessors">Non-member
 103           Accessors</a>
 104         </h5>
 105 <p>
 106           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 107           functions</a> that are generic to all distributions are supported:
 108           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 109           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 110           <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>,
 111           <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>,
 112           <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>,
 113           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 114           <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>,
 115           <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>.
 116         </p>
 117 <p>
 118           The domain of the random variable is 0 and 1, and the useful supported
 119           range is only 0 or 1.
 120         </p>
 121 <p>
 122           Outside this range, functions are undefined, or may throw domain_error
 123           exception and make an error message available.
 124         </p>
 125 <h5>
 126 <a name="math_toolkit.dist_ref.dists.bernoulli_dist.h2"></a>
 127           <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.accuracy"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.accuracy">Accuracy</a>
 128         </h5>
 129 <p>
 130           The Bernoulli distribution is implemented with simple arithmetic operators
 131           and so should have errors within an epsilon or two.
 132         </p>
 133 <h5>
 134 <a name="math_toolkit.dist_ref.dists.bernoulli_dist.h3"></a>
 135           <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.implementation"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.implementation">Implementation</a>
 136         </h5>
 137 <p>
 138           In the following table <span class="emphasis"><em>p</em></span> is the probability of success
 139           and <span class="emphasis"><em>q = 1-p</em></span>. <span class="emphasis"><em>k</em></span> is the random
 140           variate, either 0 or 1.
 141         </p>
 142 <div class="note"><table border="0" summary="Note">
 143 <tr>
 144 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 145 <th align="left">Note</th>
 146 </tr>
 147 <tr><td align="left" valign="top">
 148 <p>
 149             The Bernoulli distribution is implemented here as a <span class="emphasis"><em>strict
 150             discrete</em></span> distribution. If a generalised version, allowing
 151             k to be any real, is required then the binomial distribution with a single
 152             trial should be used, for example:
 153           </p>
 154 <p>
 155             <code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">(</span><span class="number">1</span><span class="special">,</span>
 156             <span class="number">0.25</span><span class="special">)</span></code>
 157           </p>
 158 </td></tr>
 159 </table></div>
 160 <div class="informaltable"><table class="table">
 161 <colgroup>
 162 <col>
 163 <col>
 164 </colgroup>
 165 <thead><tr>
 166 <th>
 167                   <p>
 168                     Function
 169                   </p>
 170                 </th>
 171 <th>
 172                   <p>
 173                     Implementation Notes
 174                   </p>
 175                 </th>
 176 </tr></thead>
 177 <tbody>
 178 <tr>
 179 <td>
 180                   <p>
 181                     Supported range
 182                   </p>
 183                 </td>
 184 <td>
 185                   <p>
 186                     {0, 1}
 187                   </p>
 188                 </td>
 189 </tr>
 190 <tr>
 191 <td>
 192                   <p>
 193                     pdf
 194                   </p>
 195                 </td>
 196 <td>
 197                   <p>
 198                     Using the relation: pdf = 1 - p for k = 0, else p
 199                   </p>
 200                 </td>
 201 </tr>
 202 <tr>
 203 <td>
 204                   <p>
 205                     cdf
 206                   </p>
 207                 </td>
 208 <td>
 209                   <p>
 210                     Using the relation: cdf = 1 - p for k = 0, else 1
 211                   </p>
 212                 </td>
 213 </tr>
 214 <tr>
 215 <td>
 216                   <p>
 217                     cdf complement
 218                   </p>
 219                 </td>
 220 <td>
 221                   <p>
 222                     q = 1 - p
 223                   </p>
 224                 </td>
 225 </tr>
 226 <tr>
 227 <td>
 228                   <p>
 229                     quantile
 230                   </p>
 231                 </td>
 232 <td>
 233                   <p>
 234                     if x &lt;= (1-p) 0 else 1
 235                   </p>
 236                 </td>
 237 </tr>
 238 <tr>
 239 <td>
 240                   <p>
 241                     quantile from the complement
 242                   </p>
 243                 </td>
 244 <td>
 245                   <p>
 246                     if x &lt;= (1-p) 1 else 0
 247                   </p>
 248                 </td>
 249 </tr>
 250 <tr>
 251 <td>
 252                   <p>
 253                     mean
 254                   </p>
 255                 </td>
 256 <td>
 257                   <p>
 258                     p
 259                   </p>
 260                 </td>
 261 </tr>
 262 <tr>
 263 <td>
 264                   <p>
 265                     variance
 266                   </p>
 267                 </td>
 268 <td>
 269                   <p>
 270                     p * (1 - p)
 271                   </p>
 272                 </td>
 273 </tr>
 274 <tr>
 275 <td>
 276                   <p>
 277                     mode
 278                   </p>
 279                 </td>
 280 <td>
 281                   <p>
 282                     if (p &lt; 0.5) 0 else 1
 283                   </p>
 284                 </td>
 285 </tr>
 286 <tr>
 287 <td>
 288                   <p>
 289                     skewness
 290                   </p>
 291                 </td>
 292 <td>
 293                   <p>
 294                     (1 - 2 * p) / sqrt(p * q)
 295                   </p>
 296                 </td>
 297 </tr>
 298 <tr>
 299 <td>
 300                   <p>
 301                     kurtosis
 302                   </p>
 303                 </td>
 304 <td>
 305                   <p>
 306                     6 * p * p - 6 * p +1/ p * q
 307                   </p>
 308                 </td>
 309 </tr>
 310 <tr>
 311 <td>
 312                   <p>
 313                     kurtosis excess
 314                   </p>
 315                 </td>
 316 <td>
 317                   <p>
 318                     kurtosis -3
 319                   </p>
 320                 </td>
 321 </tr>
 322 </tbody>
 323 </table></div>
 324 <h5>
 325 <a name="math_toolkit.dist_ref.dists.bernoulli_dist.h4"></a>
 326           <span class="phrase"><a name="math_toolkit.dist_ref.dists.bernoulli_dist.references"></a></span><a class="link" href="bernoulli_dist.html#math_toolkit.dist_ref.dists.bernoulli_dist.references">References</a>
 327         </h5>
 328 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 329 <li class="listitem">
 330               <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Wikpedia
 331               Bernoulli distribution</a>
 332             </li>
 333 <li class="listitem">
 334               <a href="http://mathworld.wolfram.com/BernoulliDistribution.html" target="_top">Weisstein,
 335               Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram
 336               Web Resource.</a>
 337             </li>
 338 </ul></div>
 339 </div>
 340 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 341 <td align="left"></td>
 342 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 343       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 344       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 345       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 346         Distributed under the Boost Software License, Version 1.0. (See accompanying
 347         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>)
 348       </p>
 349 </div></td>
 350 </tr></table>
 351 <hr>
 352 <div class="spirit-nav">
 353 <a accesskey="p" href="arcine_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="beta_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
 354 </div>
 355 </body>
 356 </html>