<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Bernoulli Distribution</title> <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../index.html" title="Math Toolkit"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="../dists.html" title="Distributions"> <link rel="next" href="beta_dist.html" title="Beta Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a 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</h5></div></div></div> <p> </p> <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">bernoulli</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> <p> </p> <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> <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> <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<></a> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">bernoulli_distribution</span><span class="special"><></span> <span class="identifier">bernoulli</span><span class="special">;</span> <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="Policies">Policy</a><span class="special">></span> <span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span> <span class="special">{</span> <span class="keyword">public</span><span class="special">:</span> <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> <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> <span class="comment">// Accessor function. </span> <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span> <span class="comment">// Probability of success (as a fraction). </span> <span class="special">};</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> The Bernoulli distribution is a discrete distribution of the outcome of a single trial with only two results, 0 (failure) or 1 (success), with a probability of success p. </p> <p> The Bernoulli distribution is the simplest building block on which other discrete distributions of sequences of independent Bernoulli trials can be based. </p> <p> The Bernoulli is the binomial distribution (k = 1, p) with only one trial. </p> <p> <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability density function pdf</a> f(0) = 1 - p, f(1) = p. <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative distribution function</a> D(k) = if (k == 0) 1 - p else 1. </p> <p> The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability density function pdf</a> varies with the outcome of the single trial: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/bernoulli_pdf.png" align="middle"></span> </p> <p> and the <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative distribution function</a> </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/bernoulli_cdf.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.bernoulli_dist.member_functions"></a><h5> <a name="id1015345"></a> <a class="link" href="bernoulli_dist.html#math_toolkit.dist.dist_ref.dists.bernoulli_dist.member_functions">Member Functions</a> </h5> <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> </pre> <p> Constructs a <a href="http://en.wikipedia.org/wiki/bernoulli_distribution" target="_top">bernoulli distribution</a> with success_fraction <span class="emphasis"><em>p</em></span>. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span> </pre> <p> Returns the <span class="emphasis"><em>success_fraction</em></span> parameter of this distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.bernoulli_dist.non_member_accessors"></a><h5> <a name="id1015423"></a> <a class="link" href="bernoulli_dist.html#math_toolkit.dist.dist_ref.dists.bernoulli_dist.non_member_accessors">Non-member Accessors</a> </h5> <p> All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor functions</a> that are generic to all distributions are supported: <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, <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 Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. </p> <p> The domain of the random variable is 0 and 1, and the useful supported range is only 0 or 1. </p> <p> Outside this range, functions are undefined, or may throw domain_error exception and make an error message available. </p> <a name="math_toolkit.dist.dist_ref.dists.bernoulli_dist.accuracy"></a><h5> <a name="id1015526"></a> <a class="link" href="bernoulli_dist.html#math_toolkit.dist.dist_ref.dists.bernoulli_dist.accuracy">Accuracy</a> </h5> <p> The Bernoulli distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two. </p> <a name="math_toolkit.dist.dist_ref.dists.bernoulli_dist.implementation"></a><h5> <a name="id1015548"></a> <a class="link" href="bernoulli_dist.html#math_toolkit.dist.dist_ref.dists.bernoulli_dist.implementation">Implementation</a> </h5> <p> In the following table <span class="emphasis"><em>p</em></span> is the probability of success and <span class="emphasis"><em>q = 1-p</em></span>. <span class="emphasis"><em>k</em></span> is the random variate, either 0 or 1. </p> <div class="note"><table border="0" summary="Note"> <tr> <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td> <th align="left">Note</th> </tr> <tr><td align="left" valign="top"> <p> The Bernoulli distribution is implemented here as a <span class="emphasis"><em>strict discrete</em></span> distribution. If a generalised version, allowing k to be any real, is required then the binomial distribution with a single trial should be used, for example: </p> <p> <code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span></code> </p> </td></tr> </table></div> <div class="informaltable"><table class="table"> <colgroup> <col> <col> </colgroup> <thead><tr> <th> <p> Function </p> </th> <th> <p> Implementation Notes </p> </th> </tr></thead> <tbody> <tr> <td> <p> Supported range </p> </td> <td> <p> {0, 1} </p> </td> </tr> <tr> <td> <p> pdf </p> </td> <td> <p> Using the relation: pdf = 1 - p for k = 0, else p </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relation: cdf = 1 - p for k = 0, else 1 </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> q = 1 - p </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> if x <= (1-p) 0 else 1 </p> </td> </tr> <tr> <td> <p> quantile from the complement </p> </td> <td> <p> if x <= (1-p) 1 else 0 </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> p </p> </td> </tr> <tr> <td> <p> variance </p> </td> <td> <p> p * (1 - p) </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> if (p < 0.5) 0 else 1 </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> (1 - 2 * p) / sqrt(p * q) </p> </td> </tr> <tr> <td> <p> kurtosis </p> </td> <td> <p> 6 * p * p - 6 * p +1/ p * q </p> </td> </tr> <tr> <td> <p> kurtosis excess </p> </td> <td> <p> kurtosis -3 </p> </td> </tr> </tbody> </table></div> <a name="math_toolkit.dist.dist_ref.dists.bernoulli_dist.references"></a><h5> <a name="id1015877"></a> <a class="link" href="bernoulli_dist.html#math_toolkit.dist.dist_ref.dists.bernoulli_dist.references">References</a> </h5> <div class="itemizedlist"><ul type="disc"> <li> <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Wikpedia Bernoulli distribution</a> </li> <li> <a href="../../../../" target="_top">Weisstein, Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram Web Resource.</a> </li> </ul></div> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="../dists.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> </div> </body> </html>