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<title>Binomial Distribution</title>
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<div class="section math_toolkit_dist_dist_ref_dists_binomial_dist">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist"></a><a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
Distribution</a>
</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">binomial</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">binomial_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><></span> <span class="identifier">binomial</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">binomial_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="keyword">static</span> <span class="keyword">const</span> <span class="emphasis"><em>unspecified-type</em></span> <span class="identifier">clopper_pearson_exact_interval</span><span class="special">;</span>
<span class="keyword">static</span> <span class="keyword">const</span> <span class="emphasis"><em>unspecified-type</em></span> <span class="identifier">jeffreys_prior_interval</span><span class="special">;</span>
<span class="comment">// construct:</span>
<span class="identifier">binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
<span class="comment">// parameter access::</span>
<span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
<span class="comment">// Bounds on success fraction:</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">,</span>
<span class="emphasis"><em>unspecified-type</em></span> <span class="identifier">method</span> <span class="special">=</span> <span class="identifier">clopper_pearson_exact_interval</span><span class="special">);</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">,</span>
<span class="emphasis"><em>unspecified-type</em></span> <span class="identifier">method</span> <span class="special">=</span> <span class="identifier">clopper_pearson_exact_interval</span><span class="special">);</span>
<span class="comment">// estimate min/max number of trials:</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of events</span>
<span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// risk level</span>
<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of events</span>
<span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// risk level</span>
<span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
The class type <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>
represents a <a href="http://mathworld.wolfram.com/BinomialDistribution.html" target="_top">binomial
distribution</a>: it is used when there are exactly two mutually
exclusive outcomes of a trial. These outcomes are labelled "success"
and "failure". The <a href="../../../../binomial" target="_top">distribution</a>
is used to obtain the probability of observing k successes in N trials,
with the probability of success on a single trial denoted by p. The binomial
distribution assumes that p is fixed for all trials.
</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 random variable for the binomial distribution is the number of
successes, (the number of trials is a fixed property of the distribution)
whereas for the negative binomial, the random variable is the number
of trials, for a fixed number of successes.
</p></td></tr>
</table></div>
<p>
The PDF for the binomial distribution is given by:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/binomial_ref2.png"></span>
</p>
<p>
The following two graphs illustrate how the PDF changes depending upon
the distributions parameters, first we'll keep the success fraction
<span class="emphasis"><em>p</em></span> fixed at 0.5, and vary the sample size:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/binomial_pdf_1.png" align="middle"></span>
</p>
<p>
Alternatively, we can keep the sample size fixed at N=20 and vary the
success fraction <span class="emphasis"><em>p</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/binomial_pdf_2.png" align="middle"></span>
</p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top">
<p>
The Binomial distribution is a discrete distribution: internally, functions
like the <code class="computeroutput"><span class="identifier">cdf</span></code> and <code class="computeroutput"><span class="identifier">pdf</span></code> are treated "as if"
they are continuous functions, but in reality the results returned
from these functions only have meaning if an integer value is provided
for the random variate argument.
</p>
<p>
The quantile function will by default return an integer result that
has been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower
quantiles (where the probability is less than 0.5) are rounded downward,
and upper quantiles (where the probability is greater than 0.5) are
rounded upwards. This behaviour ensures that if an X% quantile is requested,
then <span class="emphasis"><em>at least</em></span> the requested coverage will be present
in the central region, and <span class="emphasis"><em>no more than</em></span> the requested
coverage will be present in the tails.
</p>
<p>
This behaviour can be changed so that the quantile functions are rounded
differently, or even return a real-valued result using <a class="link" href="../../../policy/pol_overview.html" title="Policy Overview">Policies</a>.
It is strongly recommended that you read the tutorial <a class="link" href="../../../policy/pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
Quantiles of Discrete Distributions</a> before using the quantile
function on the Binomial distribution. The <a class="link" href="../../../policy/pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
docs</a> describe how to change the rounding policy for these distributions.
</p>
</td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h0"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.member_functions"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.member_functions">Member
Functions</a>
</h5>
<h6>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h1"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.construct"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.construct">Construct</a>
</h6>
<pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span>
</pre>
<p>
Constructor: <span class="emphasis"><em>n</em></span> is the total number of trials, <span class="emphasis"><em>p</em></span>
is the probability of success of a single trial.
</p>
<p>
Requires <code class="computeroutput"><span class="number">0</span> <span class="special"><=</span>
<span class="identifier">p</span> <span class="special"><=</span>
<span class="number">1</span></code>, and <code class="computeroutput"><span class="identifier">n</span>
<span class="special">>=</span> <span class="number">0</span></code>,
otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
</p>
<h6>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h2"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.accessors"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.accessors">Accessors</a>
</h6>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the parameter <span class="emphasis"><em>p</em></span> from which this distribution
was constructed.
</p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the parameter <span class="emphasis"><em>n</em></span> from which this distribution
was constructed.
</p>
<h6>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h3"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.lower_bound_on_the_success_fraction"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.lower_bound_on_the_success_fraction">Lower
Bound on the Success Fraction</a>
</h6>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
<span class="emphasis"><em>unspecified-type</em></span> <span class="identifier">method</span> <span class="special">=</span> <span class="identifier">clopper_pearson_exact_interval</span><span class="special">);</span>
</pre>
<p>
Returns a lower bound on the success fraction:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl>
<dt><span class="term">trials</span></dt>
<dd><p>
The total number of trials conducted.
</p></dd>
<dt><span class="term">successes</span></dt>
<dd><p>
The number of successes that occurred.
</p></dd>
<dt><span class="term">alpha</span></dt>
<dd><p>
The largest acceptable probability that the true value of the success
fraction is <span class="bold"><strong>less than</strong></span> the value
returned.
</p></dd>
<dt><span class="term">method</span></dt>
<dd><p>
An optional parameter that specifies the method to be used to compute
the interval (See below).
</p></dd>
</dl>
</div>
<p>
For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span>
trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>,
but if you want to be 95% sure that the true value is <span class="bold"><strong>greater
than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then:
</p>
<pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
<span class="identifier">n</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
</pre>
<p>
<a class="link" href="../../stat_tut/weg/binom_eg/binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution">See
worked example.</a>
</p>
<p>
There are currently two possible values available for the <span class="emphasis"><em>method</em></span>
optional parameter: <span class="emphasis"><em>clopper_pearson_exact_interval</em></span>
or <span class="emphasis"><em>jeffreys_prior_interval</em></span>. These constants are
both members of class template <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>,
so usage is for example:
</p>
<pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
<span class="identifier">n</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span>
</pre>
<p>
The default method if this parameter is not specified is the Clopper
Pearson "exact" interval. This produces an interval that guarantees
at least <code class="computeroutput"><span class="number">100</span><span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">)%</span></code> coverage, but which is known to be
overly conservative, sometimes producing intervals with much greater
than the requested coverage.
</p>
<p>
The alternative calculation method produces a non-informative Jeffreys
Prior interval. It produces <code class="computeroutput"><span class="number">100</span><span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">)%</span></code>
coverage only <span class="emphasis"><em>in the average case</em></span>, though is typically
very close to the requested coverage level. It is one of the main methods
of calculation recommended in the review by Brown, Cai and DasGupta.
</p>
<p>
Please note that the "textbook" calculation method using a
normal approximation (the Wald interval) is deliberately not provided:
it is known to produce consistently poor results, even when the sample
size is surprisingly large. Refer to Brown, Cai and DasGupta for a full
explanation. Many other methods of calculation are available, and may
be more appropriate for specific situations. Unfortunately there appears
to be no consensus amongst statisticians as to which is "best":
refer to the discussion at the end of Brown, Cai and DasGupta for examples.
</p>
<p>
The two methods provided here were chosen principally because they can
be used for both one and two sided intervals. See also:
</p>
<p>
Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001), Interval
Estimation for a Binomial Proportion, Statistical Science, Vol. 16, No.
2, 101-133.
</p>
<p>
T. Tony Cai (2005), One-sided confidence intervals in discrete distributions,
Journal of Statistical Planning and Inference 131, 63-88.
</p>
<p>
Agresti, A. and Coull, B. A. (1998). Approximate is better than "exact"
for interval estimation of binomial proportions. Amer. Statist. 52 119-126.
</p>
<p>
Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial
limits illustrated in the case of the binomial. Biometrika 26 404-413.
</p>
<h6>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h4"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.upper_bound_on_the_success_fraction"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.upper_bound_on_the_success_fraction">Upper
Bound on the Success Fraction</a>
</h6>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
<span class="emphasis"><em>unspecified-type</em></span> <span class="identifier">method</span> <span class="special">=</span> <span class="identifier">clopper_pearson_exact_interval</span><span class="special">);</span>
</pre>
<p>
Returns an upper bound on the success fraction:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl>
<dt><span class="term">trials</span></dt>
<dd><p>
The total number of trials conducted.
</p></dd>
<dt><span class="term">successes</span></dt>
<dd><p>
The number of successes that occurred.
</p></dd>
<dt><span class="term">alpha</span></dt>
<dd><p>
The largest acceptable probability that the true value of the success
fraction is <span class="bold"><strong>greater than</strong></span> the value
returned.
</p></dd>
<dt><span class="term">method</span></dt>
<dd><p>
An optional parameter that specifies the method to be used to compute
the interval. Refer to the documentation for <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
above for the meaning of the method options.
</p></dd>
</dl>
</div>
<p>
For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span>
trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>,
but if you want to be 95% sure that the true value is <span class="bold"><strong>less
than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then:
</p>
<pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
<span class="identifier">n</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
</pre>
<p>
<a class="link" href="../../stat_tut/weg/binom_eg/binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution">See
worked example.</a>
</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>
In order to obtain a two sided bound on the success fraction, you call
both <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
<span class="bold"><strong>and</strong></span> <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
each with the same arguments.
</p>
<p>
If the desired risk level that the true success fraction lies outside
the bounds is α, then you pass α/2 to these functions.
</p>
<p>
So for example a two sided 95% confidence interval would be obtained
by passing α = 0.025 to each of the functions.
</p>
<p>
<a class="link" href="../../stat_tut/weg/binom_eg/binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution">See
worked example.</a>
</p>
</td></tr>
</table></div>
<h6>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h5"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.estimating_the_number_of_trials_required_for_a_certain_number_of_successes"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.estimating_the_number_of_trials_required_for_a_certain_number_of_successes">Estimating
the Number of Trials Required for a Certain Number of Successes</a>
</h6>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of events</span>
<span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold</span>
</pre>
<p>
This function estimates the minimum number of trials required to ensure
that more than k events is observed with a level of risk <span class="emphasis"><em>alpha</em></span>
that k or fewer events occur.
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl>
<dt><span class="term">k</span></dt>
<dd><p>
The number of success observed.
</p></dd>
<dt><span class="term">p</span></dt>
<dd><p>
The probability of success for each trial.
</p></dd>
<dt><span class="term">alpha</span></dt>
<dd><p>
The maximum acceptable probability that k events or fewer will
be observed.
</p></dd>
</dl>
</div>
<p>
For example:
</p>
<pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
</pre>
<p>
Returns the smallest number of trials we must conduct to be 95% sure
of seeing 10 events that occur with frequency one half.
</p>
<h6>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h6"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.estimating_the_maximum_number_of_trials_to_ensure_no_more_than_a_certain_number_of_successes"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.estimating_the_maximum_number_of_trials_to_ensure_no_more_than_a_certain_number_of_successes">Estimating
the Maximum Number of Trials to Ensure no more than a Certain Number
of Successes</a>
</h6>
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
<span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of events</span>
<span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction</span>
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold</span>
</pre>
<p>
This function estimates the maximum number of trials we can conduct to
ensure that k successes or fewer are observed, with a risk <span class="emphasis"><em>alpha</em></span>
that more than k occur.
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl>
<dt><span class="term">k</span></dt>
<dd><p>
The number of success observed.
</p></dd>
<dt><span class="term">p</span></dt>
<dd><p>
The probability of success for each trial.
</p></dd>
<dt><span class="term">alpha</span></dt>
<dd><p>
The maximum acceptable probability that more than k events will
be observed.
</p></dd>
</dl>
</div>
<p>
For example:
</p>
<pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1e-6</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span>
</pre>
<p>
Returns the largest number of trials we can conduct and still be 95%
certain of not observing any events that occur with one in a million
frequency. This is typically used in failure analysis.
</p>
<p>
<a class="link" href="../../stat_tut/weg/binom_eg/binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution.">See
Worked Example.</a>
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h7"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.non_member_accessors"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_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 for the random variable <span class="emphasis"><em>k</em></span> is <code class="computeroutput"><span class="number">0</span> <span class="special"><=</span> <span class="identifier">k</span> <span class="special"><=</span> <span class="identifier">N</span></code>, otherwise a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
is returned.
</p>
<p>
It's worth taking a moment to define what these accessors actually mean
in the context of this distribution:
</p>
<div class="table">
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.meaning_of_the_non_member_accessors"></a><p class="title"><b>Table 13. Meaning of the non-member accessors</b></p>
<div class="table-contents"><table class="table" summary="Meaning of the non-member accessors">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
Meaning
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
<a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>
</p>
</td>
<td>
<p>
The probability of obtaining <span class="bold"><strong>exactly
k successes</strong></span> from n trials with success fraction
p. For example:
</p>
<p>
<code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
<span class="identifier">p</span><span class="special">),</span>
<span class="identifier">k</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>
</p>
</td>
<td>
<p>
The probability of obtaining <span class="bold"><strong>k successes
or fewer</strong></span> from n trials with success fraction p.
For example:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
<span class="identifier">p</span><span class="special">),</span>
<span class="identifier">k</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../nmp.html#math.dist.ccdf">Complement of the Cumulative
Distribution Function</a>
</p>
</td>
<td>
<p>
The probability of obtaining <span class="bold"><strong>more than
k successes</strong></span> from n trials with success fraction
p. For example:
</p>
<p>
<code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
<span class="identifier">p</span><span class="special">),</span>
<span class="identifier">k</span><span class="special">))</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>
</p>
</td>
<td>
<p>
The <span class="bold"><strong>greatest</strong></span> number of successes
that may be observed from n trials with success fraction p,
at probability P. Note that the value returned is a real-number,
and not an integer. Depending on the use case you may want
to take either the floor or ceiling of the result. For example:
</p>
<p>
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
<span class="identifier">p</span><span class="special">),</span>
<span class="identifier">P</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../nmp.html#math.dist.quantile_c">Quantile from the complement
of the probability</a>
</p>
</td>
<td>
<p>
The <span class="bold"><strong>smallest</strong></span> number of successes
that may be observed from n trials with success fraction p,
at probability P. Note that the value returned is a real-number,
and not an integer. Depending on the use case you may want
to take either the floor or ceiling of the result. For example:
</p>
<p>
<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span>
<span class="identifier">p</span><span class="special">),</span>
<span class="identifier">P</span><span class="special">))</span></code>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h8"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.examples"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.examples">Examples</a>
</h5>
<p>
Various <a class="link" href="../../stat_tut/weg/binom_eg.html" title="Binomial Distribution Examples">worked
examples</a> are available illustrating the use of the binomial distribution.
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h9"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.accuracy"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.accuracy">Accuracy</a>
</h5>
<p>
This distribution is implemented using the incomplete beta functions
<a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
and <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>,
please refer to these functions for information on accuracy.
</p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h10"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.implementation"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.implementation">Implementation</a>
</h5>
<p>
In the following table <span class="emphasis"><em>p</em></span> is the probability that
one trial will be successful (the success fraction), <span class="emphasis"><em>n</em></span>
is the number of trials, <span class="emphasis"><em>k</em></span> is the number of successes,
<span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
</p>
<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>
pdf
</p>
</td>
<td>
<p>
Implementation is in terms of <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>:
if <sub>n</sub>C<sub>k </sub> is the binomial coefficient of a and b, then we have:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../equations/binomial_ref1.png"></span>
</p>
<p>
Which can be evaluated as <code class="computeroutput"><span class="identifier">ibeta_derivative</span><span class="special">(</span><span class="identifier">k</span><span class="special">+</span><span class="number">1</span><span class="special">,</span> <span class="identifier">n</span><span class="special">-</span><span class="identifier">k</span><span class="special">+</span><span class="number">1</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span>
<span class="special">(</span><span class="identifier">n</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code>
</p>
<p>
The function <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>
is used here, since it has already been optimised for the lowest
possible error - indeed this is really just a thin wrapper
around part of the internals of the incomplete beta function.
</p>
<p>
There are also various special cases: refer to the code for
details.
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<p>
Using the relation:
</p>
<p>
</p>
<pre xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="table-programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">I</span><span class="special">[</span><span class="identifier">sub</span> <span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">](</span><span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">k</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span>
<span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">I</span><span class="special">[</span><span class="identifier">sub</span> <span class="identifier">p</span><span class="special">](</span><span class="identifier">k</span> <span class="special">+</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">)</span>
<span class="special">=</span> <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a><span class="special">(</span><span class="identifier">k</span> <span class="special">+</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span></pre>
<p>
</p>
<p>
There are also various special cases: refer to the code for
details.
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf complement
</p>
</td>
<td>
<p>
Using the relation: q = <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(k
+ 1, n - k, p)
</p>
<p>
There are also various special cases: refer to the code for
details.
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Since the cdf is non-linear in variate <span class="emphasis"><em>k</em></span>
none of the inverse incomplete beta functions can be used here.
Instead the quantile is found numerically using a derivative
free method (<a class="link" href="../../../toolkit/internals1/roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">TOMS
Algorithm 748</a>).
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
Found numerically as above.
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">p</span> <span class="special">*</span>
<span class="identifier">n</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">p</span> <span class="special">*</span>
<span class="identifier">n</span> <span class="special">*</span>
<span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">p</span> <span class="special">*</span>
<span class="special">(</span><span class="identifier">n</span>
<span class="special">+</span> <span class="number">1</span><span class="special">))</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="special">(</span><span class="number">1</span>
<span class="special">-</span> <span class="number">2</span>
<span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span>
<span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">n</span> <span class="special">*</span>
<span class="identifier">p</span> <span class="special">*</span>
<span class="special">(</span><span class="number">1</span>
<span class="special">-</span> <span class="identifier">p</span><span class="special">))</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="number">3</span> <span class="special">-</span>
<span class="special">(</span><span class="number">6</span>
<span class="special">/</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">+</span>
<span class="special">(</span><span class="number">1</span>
<span class="special">/</span> <span class="special">(</span><span class="identifier">n</span> <span class="special">*</span>
<span class="identifier">p</span> <span class="special">*</span>
<span class="special">(</span><span class="number">1</span>
<span class="special">-</span> <span class="identifier">p</span><span class="special">)))</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis excess
</p>
</td>
<td>
<p>
<code class="computeroutput"><span class="special">(</span><span class="number">1</span>
<span class="special">-</span> <span class="number">6</span>
<span class="special">*</span> <span class="identifier">p</span>
<span class="special">*</span> <span class="identifier">q</span><span class="special">)</span> <span class="special">/</span>
<span class="special">(</span><span class="identifier">n</span>
<span class="special">*</span> <span class="identifier">p</span>
<span class="special">*</span> <span class="identifier">q</span><span class="special">)</span></code>
</p>
</td>
</tr>
<tr>
<td>
<p>
parameter estimation
</p>
</td>
<td>
<p>
The member functions <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
<code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">find_number_of_trials</span></code>
are implemented in terms of the inverse incomplete beta functions
<a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>,
<a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>,
and <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_invb</a>
respectively
</p>
</td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.binomial_dist.h11"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.binomial_dist.references"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist.dist_ref.dists.binomial_dist.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" type="disc">
<li class="listitem">
<a href="http://mathworld.wolfram.com/BinomialDistribution.html" target="_top">Weisstein,
Eric W. "Binomial Distribution." From MathWorld--A Wolfram
Web Resource</a>.
</li>
<li class="listitem">
<a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia
binomial distribution</a>.
</li>
<li class="listitem">
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm" target="_top">NIST
Explorary Data Analysis</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 © 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<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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