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| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist_ref.dists.binomial_dist"></a><a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial |
| Distribution</a> |
| </h4></div></div></div> |
| <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> |
| <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="Chapter 14. 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> |
| <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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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 class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial |
| 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.svg"></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.svg" 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.svg" 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="../../pol_overview.html" title="Policy Overview">Policies</a>. |
| It is strongly recommended that you read the tutorial <a class="link" href="../../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="../../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_ref.dists.binomial_dist.h0"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.member_functions"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.binomial_dist.h1"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.construct"></a></span><a class="link" href="binomial_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. |
| </p> |
| <h6> |
| <a name="math_toolkit.dist_ref.dists.binomial_dist.h2"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.accessors"></a></span><a class="link" href="binomial_dist.html#math_toolkit.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_ref.dists.binomial_dist.h3"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.lower_bound_on_the_success_fract"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.lower_bound_on_the_success_fract">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 class="variablelist"> |
| <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_ref.dists.binomial_dist.h4"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.upper_bound_on_the_success_fract"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.upper_bound_on_the_success_fract">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 class="variablelist"> |
| <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_ref.dists.binomial_dist.h5"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.estimating_the_number_of_trials_"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.estimating_the_number_of_trials_">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 class="variablelist"> |
| <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_ref.dists.binomial_dist.h6"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.estimating_the_maximum_number_of"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.estimating_the_maximum_number_of">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 class="variablelist"> |
| <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_ref.dists.binomial_dist.h7"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.non_member_accessors"></a></span><a class="link" href="binomial_dist.html#math_toolkit.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_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 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="../../error_handling.html#math_toolkit.error_handling.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_ref.dists.binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table 5.1. 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_toolkit.dist_ref.nmp.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_toolkit.dist_ref.nmp.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_toolkit.dist_ref.nmp.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_toolkit.dist_ref.nmp.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_toolkit.dist_ref.nmp.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_ref.dists.binomial_dist.h8"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.examples"></a></span><a class="link" href="binomial_dist.html#math_toolkit.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_ref.dists.binomial_dist.h9"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.accuracy"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| This distribution is implemented using the incomplete beta functions <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> and <a class="link" href="../../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_ref.dists.binomial_dist.h10"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.implementation"></a></span><a class="link" href="binomial_dist.html#math_toolkit.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="../../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.svg"></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="../../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> |
| <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="../../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> |
| 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="../../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="../../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="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>, |
| <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>, |
| and <a class="link" href="../../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_ref.dists.binomial_dist.h11"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.binomial_dist.references"></a></span><a class="link" href="binomial_dist.html#math_toolkit.dist_ref.dists.binomial_dist.references">References</a> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-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, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| 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></td> |
| </tr></table> |
| <hr> |
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