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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.bernoulli_dist"></a><a class="link" href="bernoulli_dist.html" title="Bernoulli Distribution">Bernoulli
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Distribution</a>
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</h4></div></div></div>
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<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>
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<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>
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<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>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. 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>
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<span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">bernoulli_distribution</span><span class="special"><></span> <span class="identifier">bernoulli</span><span class="special">;</span>
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<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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">bernoulli_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
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<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>
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<span class="comment">// Accessor function.</span>
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<span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span>
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<span class="comment">// Probability of success (as a fraction).</span>
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<span class="special">};</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The Bernoulli distribution is a discrete distribution of the outcome of
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a single trial with only two results, 0 (failure) or 1 (success), with
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a probability of success p.
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</p>
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<p>
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The Bernoulli distribution is the simplest building block on which other
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discrete distributions of sequences of independent Bernoulli trials can
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be based.
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</p>
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<p>
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The Bernoulli is the binomial distribution (k = 1, p) with only one trial.
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</p>
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<p>
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<a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
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density function pdf</a>
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">f(0) = 1 - p, f(1) = p</span>
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</p></blockquote></div>
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<p>
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<a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative
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distribution function</a>
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">D(k) = if (k == 0) 1 - p else 1</span>
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</p></blockquote></div>
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<p>
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The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
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density function pdf</a> varies with the outcome of the single trial:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_pdf.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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and the <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative
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distribution function</a>
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_cdf.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h0"></a>
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<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
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Functions</a>
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</h5>
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<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>
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</pre>
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<p>
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Constructs a <a href="http://en.wikipedia.org/wiki/bernoulli_distribution" target="_top">bernoulli
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distribution</a> with success_fraction <span class="emphasis"><em>p</em></span>.
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</p>
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<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span>
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</pre>
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<p>
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Returns the <span class="emphasis"><em>success_fraction</em></span> parameter of this distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h1"></a>
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<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
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Accessors</a>
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</h5>
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<p>
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All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
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functions</a> that are generic to all distributions are supported:
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
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<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>,
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<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>,
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<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>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
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<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>,
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<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>.
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</p>
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<p>
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The domain of the random variable is 0 and 1, and the useful supported
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range is only 0 or 1.
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</p>
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<p>
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Outside this range, functions are undefined, or may throw domain_error
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exception and make an error message available.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h2"></a>
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<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>
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</h5>
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<p>
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The Bernoulli distribution is implemented with simple arithmetic operators
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and so should have errors within an epsilon or two.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h3"></a>
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<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>
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</h5>
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<p>
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In the following table <span class="emphasis"><em>p</em></span> is the probability of success
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and <span class="emphasis"><em>q = 1-p</em></span>. <span class="emphasis"><em>k</em></span> is the random
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variate, either 0 or 1.
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</p>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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The Bernoulli distribution is implemented here as a <span class="emphasis"><em>strict
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discrete</em></span> distribution. If a generalised version, allowing
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k to be any real, is required then the binomial distribution with a single
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trial should be used, for example:
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">(</span><span class="number">1</span><span class="special">,</span>
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<span class="number">0.25</span><span class="special">)</span></code>
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</p>
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</td></tr>
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</table></div>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Implementation Notes
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Supported range
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</p>
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</td>
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<td>
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<p>
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{0, 1}
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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pdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: pdf = 1 - p for k = 0, else p
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: cdf = 1 - p for k = 0, else 1
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf complement
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</p>
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</td>
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<td>
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<p>
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q = 1 - p
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile
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</p>
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</td>
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<td>
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<p>
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if x <= (1-p) 0 else 1
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile from the complement
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</p>
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</td>
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<td>
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<p>
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if x <= (1-p) 1 else 0
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mean
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</p>
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</td>
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<td>
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<p>
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p
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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variance
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</p>
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</td>
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<td>
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<p>
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p * (1 - p)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mode
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</p>
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</td>
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<td>
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<p>
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if (p < 0.5) 0 else 1
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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skewness
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</p>
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</td>
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<td>
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<p>
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(1 - 2 * p) / sqrt(p * q)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis
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</p>
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</td>
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<td>
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<p>
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6 * p * p - 6 * p +1/ p * q
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis excess
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</p>
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</td>
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<td>
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<p>
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kurtosis -3
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</p>
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||
</td>
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||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<h5>
|
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<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h4"></a>
|
||
<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>
|
||
</h5>
|
||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||
<li class="listitem">
|
||
<a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Wikipedia
|
||
Bernoulli distribution</a>
|
||
</li>
|
||
<li class="listitem">
|
||
<a href="http://mathworld.wolfram.com/BernoulliDistribution.html" target="_top">Weisstein,
|
||
Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram
|
||
Web Resource.</a>
|
||
</li>
|
||
</ul></div>
|
||
</div>
|
||
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||
Walker and Xiaogang Zhang<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>
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<hr>
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<div class="spirit-nav">
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<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>
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