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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.bernoulli_dist"></a><a class="link" href="bernoulli_dist.html" title="Bernoulli Distribution">Bernoulli
Distribution</a>
</h4></div></div></div>
<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>
<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">&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>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;18.&#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>
<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">&lt;&gt;</span> <span class="identifier">bernoulli</span><span class="special">;</span>
<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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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.svg" 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.svg" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h0"></a>
<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
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>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h1"></a>
<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
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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<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>,
<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>,
<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>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<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>,
<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>.
</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>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h2"></a>
<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>
</h5>
<p>
The Bernoulli distribution is implemented with simple arithmetic operators
and so should have errors within an epsilon or two.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.bernoulli_dist.h3"></a>
<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>
</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 &lt;= (1-p) 0 else 1
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
if x &lt;= (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 &lt; 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>
<h5>
<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">Wikpedia
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>
<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 &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, 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>
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