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  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.binomial_dist"></a><a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
  28         Distribution</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <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>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">binomial_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">binomial</span><span class="special">;</span>
  38
  39 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  40 <span class="keyword">class</span> <span class="identifier">binomial_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  45
  46    <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>
  47    <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>
  48
  49    <span class="comment">// construct:</span>
  50    <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>
  51
  52    <span class="comment">// parameter access::</span>
  53    <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  54    <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  55
  56    <span class="comment">// Bounds on success fraction:</span>
  57    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
  58       <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
  59       <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
  60       <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">,</span>
  61       <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>
  62    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
  63       <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
  64       <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
  65       <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">,</span>
  66       <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>
  67
  68    <span class="comment">// estimate min/max number of trials:</span>
  69    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span>
  70       <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// number of events</span>
  71       <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// success fraction</span>
  72       <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// risk level</span>
  73
  74    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
  75       <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// number of events</span>
  76       <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// success fraction</span>
  77       <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// risk level</span>
  78 <span class="special">};</span>
  79
  80 <span class="special">}}</span> <span class="comment">// namespaces</span>
  81 </pre>
  82 <p>
  83           The class type <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>
  84           represents a <a href="http://mathworld.wolfram.com/BinomialDistribution.html" target="_top">binomial
  85           distribution</a>: it is used when there are exactly two mutually exclusive
  86           outcomes of a trial. These outcomes are labelled "success" and
  87           "failure". The <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
  88           Distribution</a> is used to obtain the probability of observing k successes
  89           in N trials, with the probability of success on a single trial denoted
  90           by p. The binomial distribution assumes that p is fixed for all trials.
  91         </p>
  92 <div class="note"><table border="0" summary="Note">
  93 <tr>
  94 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  95 <th align="left">Note</th>
  96 </tr>
  97 <tr><td align="left" valign="top"><p>
  98             The random variable for the binomial distribution is the number of successes,
  99             (the number of trials is a fixed property of the distribution) whereas
 100             for the negative binomial, the random variable is the number of trials,
 101             for a fixed number of successes.
 102           </p></td></tr>
 103 </table></div>
 104 <p>
 105           The PDF for the binomial distribution is given by:
 106         </p>
 107 <p>
 108           <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref2.svg"></span>
 109         </p>
 110 <p>
 111           The following two graphs illustrate how the PDF changes depending upon
 112           the distributions parameters, first we'll keep the success fraction <span class="emphasis"><em>p</em></span>
 113           fixed at 0.5, and vary the sample size:
 114         </p>
 115 <p>
 116           <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_1.svg" align="middle"></span>
 117         </p>
 118 <p>
 119           Alternatively, we can keep the sample size fixed at N=20 and vary the success
 120           fraction <span class="emphasis"><em>p</em></span>:
 121         </p>
 122 <p>
 123           <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_2.svg" align="middle"></span>
 124         </p>
 125 <div class="caution"><table border="0" summary="Caution">
 126 <tr>
 127 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
 128 <th align="left">Caution</th>
 129 </tr>
 130 <tr><td align="left" valign="top">
 131 <p>
 132             The Binomial distribution is a discrete distribution: internally, functions
 133             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
 134             are continuous functions, but in reality the results returned from these
 135             functions only have meaning if an integer value is provided for the random
 136             variate argument.
 137           </p>
 138 <p>
 139             The quantile function will by default return an integer result that has
 140             been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
 141             (where the probability is less than 0.5) are rounded downward, and upper
 142             quantiles (where the probability is greater than 0.5) are rounded upwards.
 143             This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
 144             least</em></span> the requested coverage will be present in the central
 145             region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
 146             will be present in the tails.
 147           </p>
 148 <p>
 149             This behaviour can be changed so that the quantile functions are rounded
 150             differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
 151             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
 152             Quantiles of Discrete Distributions</a> before using the quantile
 153             function on the Binomial distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
 154             docs</a> describe how to change the rounding policy for these distributions.
 155           </p>
 156 </td></tr>
 157 </table></div>
 158 <h5>
 159 <a name="math_toolkit.dist_ref.dists.binomial_dist.h0"></a>
 160           <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
 161           Functions</a>
 162         </h5>
 163 <h6>
 164 <a name="math_toolkit.dist_ref.dists.binomial_dist.h1"></a>
 165           <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>
 166         </h6>
 167 <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>
 168 </pre>
 169 <p>
 170           Constructor: <span class="emphasis"><em>n</em></span> is the total number of trials, <span class="emphasis"><em>p</em></span>
 171           is the probability of success of a single trial.
 172         </p>
 173 <p>
 174           Requires <code class="computeroutput"><span class="number">0</span> <span class="special">&lt;=</span>
 175           <span class="identifier">p</span> <span class="special">&lt;=</span>
 176           <span class="number">1</span></code>, and <code class="computeroutput"><span class="identifier">n</span>
 177           <span class="special">&gt;=</span> <span class="number">0</span></code>,
 178           otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 179         </p>
 180 <h6>
 181 <a name="math_toolkit.dist_ref.dists.binomial_dist.h2"></a>
 182           <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>
 183         </h6>
 184 <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>
 185 </pre>
 186 <p>
 187           Returns the parameter <span class="emphasis"><em>p</em></span> from which this distribution
 188           was constructed.
 189         </p>
 190 <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>
 191 </pre>
 192 <p>
 193           Returns the parameter <span class="emphasis"><em>n</em></span> from which this distribution
 194           was constructed.
 195         </p>
 196 <h6>
 197 <a name="math_toolkit.dist_ref.dists.binomial_dist.h3"></a>
 198           <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
 199           Bound on the Success Fraction</a>
 200         </h6>
 201 <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>
 202    <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
 203    <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
 204    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
 205    <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>
 206 </pre>
 207 <p>
 208           Returns a lower bound on the success fraction:
 209         </p>
 210 <div class="variablelist">
 211 <p class="title"><b></b></p>
 212 <dl class="variablelist">
 213 <dt><span class="term">trials</span></dt>
 214 <dd><p>
 215                 The total number of trials conducted.
 216               </p></dd>
 217 <dt><span class="term">successes</span></dt>
 218 <dd><p>
 219                 The number of successes that occurred.
 220               </p></dd>
 221 <dt><span class="term">alpha</span></dt>
 222 <dd><p>
 223                 The largest acceptable probability that the true value of the success
 224                 fraction is <span class="bold"><strong>less than</strong></span> the value
 225                 returned.
 226               </p></dd>
 227 <dt><span class="term">method</span></dt>
 228 <dd><p>
 229                 An optional parameter that specifies the method to be used to compute
 230                 the interval (See below).
 231               </p></dd>
 232 </dl>
 233 </div>
 234 <p>
 235           For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span>
 236           trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>,
 237           but if you want to be 95% sure that the true value is <span class="bold"><strong>greater
 238           than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then:
 239         </p>
 240 <pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
 241                     <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>
 242 </pre>
 243 <p>
 244           <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
 245           example.</a>
 246         </p>
 247 <p>
 248           There are currently two possible values available for the <span class="emphasis"><em>method</em></span>
 249           optional parameter: <span class="emphasis"><em>clopper_pearson_exact_interval</em></span>
 250           or <span class="emphasis"><em>jeffreys_prior_interval</em></span>. These constants are both
 251           members of class template <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>,
 252           so usage is for example:
 253         </p>
 254 <pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span>
 255     <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">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span>
 256 </pre>
 257 <p>
 258           The default method if this parameter is not specified is the Clopper Pearson
 259           "exact" interval. This produces an interval that guarantees at
 260           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
 261           conservative, sometimes producing intervals with much greater than the
 262           requested coverage.
 263         </p>
 264 <p>
 265           The alternative calculation method produces a non-informative Jeffreys
 266           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>
 267           coverage only <span class="emphasis"><em>in the average case</em></span>, though is typically
 268           very close to the requested coverage level. It is one of the main methods
 269           of calculation recommended in the review by Brown, Cai and DasGupta.
 270         </p>
 271 <p>
 272           Please note that the "textbook" calculation method using a normal
 273           approximation (the Wald interval) is deliberately not provided: it is known
 274           to produce consistently poor results, even when the sample size is surprisingly
 275           large. Refer to Brown, Cai and DasGupta for a full explanation. Many other
 276           methods of calculation are available, and may be more appropriate for specific
 277           situations. Unfortunately there appears to be no consensus amongst statisticians
 278           as to which is "best": refer to the discussion at the end of
 279           Brown, Cai and DasGupta for examples.
 280         </p>
 281 <p>
 282           The two methods provided here were chosen principally because they can
 283           be used for both one and two sided intervals. See also:
 284         </p>
 285 <p>
 286           Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001), Interval Estimation
 287           for a Binomial Proportion, Statistical Science, Vol. 16, No. 2, 101-133.
 288         </p>
 289 <p>
 290           T. Tony Cai (2005), One-sided confidence intervals in discrete distributions,
 291           Journal of Statistical Planning and Inference 131, 63-88.
 292         </p>
 293 <p>
 294           Agresti, A. and Coull, B. A. (1998). Approximate is better than "exact"
 295           for interval estimation of binomial proportions. Amer. Statist. 52 119-126.
 296         </p>
 297 <p>
 298           Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial
 299           limits illustrated in the case of the binomial. Biometrika 26 404-413.
 300         </p>
 301 <h6>
 302 <a name="math_toolkit.dist_ref.dists.binomial_dist.h4"></a>
 303           <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
 304           Bound on the Success Fraction</a>
 305         </h6>
 306 <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>
 307    <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span>
 308    <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span>
 309    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
 310    <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>
 311 </pre>
 312 <p>
 313           Returns an upper bound on the success fraction:
 314         </p>
 315 <div class="variablelist">
 316 <p class="title"><b></b></p>
 317 <dl class="variablelist">
 318 <dt><span class="term">trials</span></dt>
 319 <dd><p>
 320                 The total number of trials conducted.
 321               </p></dd>
 322 <dt><span class="term">successes</span></dt>
 323 <dd><p>
 324                 The number of successes that occurred.
 325               </p></dd>
 326 <dt><span class="term">alpha</span></dt>
 327 <dd><p>
 328                 The largest acceptable probability that the true value of the success
 329                 fraction is <span class="bold"><strong>greater than</strong></span> the value
 330                 returned.
 331               </p></dd>
 332 <dt><span class="term">method</span></dt>
 333 <dd><p>
 334                 An optional parameter that specifies the method to be used to compute
 335                 the interval. Refer to the documentation for <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
 336                 above for the meaning of the method options.
 337               </p></dd>
 338 </dl>
 339 </div>
 340 <p>
 341           For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span>
 342           trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>,
 343           but if you want to be 95% sure that the true value is <span class="bold"><strong>less
 344           than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then:
 345         </p>
 346 <pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span>
 347                     <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>
 348 </pre>
 349 <p>
 350           <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
 351           example.</a>
 352         </p>
 353 <div class="note"><table border="0" summary="Note">
 354 <tr>
 355 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 356 <th align="left">Note</th>
 357 </tr>
 358 <tr><td align="left" valign="top">
 359 <p>
 360             In order to obtain a two sided bound on the success fraction, you call
 361             both <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
 362             <span class="bold"><strong>and</strong></span> <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
 363             each with the same arguments.
 364           </p>
 365 <p>
 366             If the desired risk level that the true success fraction lies outside
 367             the bounds is &#945;, then you pass &#945;/2 to these functions.
 368           </p>
 369 <p>
 370             So for example a two sided 95% confidence interval would be obtained
 371             by passing &#945; = 0.025 to each of the functions.
 372           </p>
 373 <p>
 374             <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
 375             example.</a>
 376           </p>
 377 </td></tr>
 378 </table></div>
 379 <h6>
 380 <a name="math_toolkit.dist_ref.dists.binomial_dist.h5"></a>
 381           <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
 382           the Number of Trials Required for a Certain Number of Successes</a>
 383         </h6>
 384 <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>
 385    <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// number of events</span>
 386    <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// success fraction</span>
 387    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold</span>
 388 </pre>
 389 <p>
 390           This function estimates the minimum number of trials required to ensure
 391           that more than k events is observed with a level of risk <span class="emphasis"><em>alpha</em></span>
 392           that k or fewer events occur.
 393         </p>
 394 <div class="variablelist">
 395 <p class="title"><b></b></p>
 396 <dl class="variablelist">
 397 <dt><span class="term">k</span></dt>
 398 <dd><p>
 399                 The number of success observed.
 400               </p></dd>
 401 <dt><span class="term">p</span></dt>
 402 <dd><p>
 403                 The probability of success for each trial.
 404               </p></dd>
 405 <dt><span class="term">alpha</span></dt>
 406 <dd><p>
 407                 The maximum acceptable probability that k events or fewer will be
 408                 observed.
 409               </p></dd>
 410 </dl>
 411 </div>
 412 <p>
 413           For example:
 414         </p>
 415 <pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</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>
 416 </pre>
 417 <p>
 418           Returns the smallest number of trials we must conduct to be 95% sure of
 419           seeing 10 events that occur with frequency one half.
 420         </p>
 421 <h6>
 422 <a name="math_toolkit.dist_ref.dists.binomial_dist.h6"></a>
 423           <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
 424           the Maximum Number of Trials to Ensure no more than a Certain Number of
 425           Successes</a>
 426         </h6>
 427 <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>
 428    <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span>     <span class="comment">// number of events</span>
 429    <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span>     <span class="comment">// success fraction</span>
 430    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold</span>
 431 </pre>
 432 <p>
 433           This function estimates the maximum number of trials we can conduct to
 434           ensure that k successes or fewer are observed, with a risk <span class="emphasis"><em>alpha</em></span>
 435           that more than k occur.
 436         </p>
 437 <div class="variablelist">
 438 <p class="title"><b></b></p>
 439 <dl class="variablelist">
 440 <dt><span class="term">k</span></dt>
 441 <dd><p>
 442                 The number of success observed.
 443               </p></dd>
 444 <dt><span class="term">p</span></dt>
 445 <dd><p>
 446                 The probability of success for each trial.
 447               </p></dd>
 448 <dt><span class="term">alpha</span></dt>
 449 <dd><p>
 450                 The maximum acceptable probability that more than k events will be
 451                 observed.
 452               </p></dd>
 453 </dl>
 454 </div>
 455 <p>
 456           For example:
 457         </p>
 458 <pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</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>
 459 </pre>
 460 <p>
 461           Returns the largest number of trials we can conduct and still be 95% certain
 462           of not observing any events that occur with one in a million frequency.
 463           This is typically used in failure analysis.
 464         </p>
 465 <p>
 466           <a class="link" href="../../stat_tut/weg/binom_eg/binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution.">See Worked
 467           Example.</a>
 468         </p>
 469 <h5>
 470 <a name="math_toolkit.dist_ref.dists.binomial_dist.h7"></a>
 471           <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
 472           Accessors</a>
 473         </h5>
 474 <p>
 475           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 476           functions</a> that are generic to all distributions are supported:
 477           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 478           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 479           <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>,
 480           <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>,
 481           <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>,
 482           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 483           <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>,
 484           <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>.
 485         </p>
 486 <p>
 487           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">&lt;=</span> <span class="identifier">k</span> <span class="special">&lt;=</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>
 488           is returned.
 489         </p>
 490 <p>
 491           It's worth taking a moment to define what these accessors actually mean
 492           in the context of this distribution:
 493         </p>
 494 <div class="table">
 495 <a name="math_toolkit.dist_ref.dists.binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table&#160;5.1.&#160;Meaning of the non-member accessors</b></p>
 496 <div class="table-contents"><table class="table" summary="Meaning of the non-member accessors">
 497 <colgroup>
 498 <col>
 499 <col>
 500 </colgroup>
 501 <thead><tr>
 502 <th>
 503                   <p>
 504                     Function
 505                   </p>
 506                 </th>
 507 <th>
 508                   <p>
 509                     Meaning
 510                   </p>
 511                 </th>
 512 </tr></thead>
 513 <tbody>
 514 <tr>
 515 <td>
 516                   <p>
 517                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
 518                     Function</a>
 519                   </p>
 520                 </td>
 521 <td>
 522                   <p>
 523                     The probability of obtaining <span class="bold"><strong>exactly k
 524                     successes</strong></span> from n trials with success fraction p. For
 525                     example:
 526                   </p>
 527                   <p>
 528                     <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>
 529                     <span class="identifier">p</span><span class="special">),</span>
 530                     <span class="identifier">k</span><span class="special">)</span></code>
 531                   </p>
 532                 </td>
 533 </tr>
 534 <tr>
 535 <td>
 536                   <p>
 537                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
 538                     Function</a>
 539                   </p>
 540                 </td>
 541 <td>
 542                   <p>
 543                     The probability of obtaining <span class="bold"><strong>k successes
 544                     or fewer</strong></span> from n trials with success fraction p. For
 545                     example:
 546                   </p>
 547                   <p>
 548                     <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>
 549                     <span class="identifier">p</span><span class="special">),</span>
 550                     <span class="identifier">k</span><span class="special">)</span></code>
 551                   </p>
 552                 </td>
 553 </tr>
 554 <tr>
 555 <td>
 556                   <p>
 557                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of
 558                     the Cumulative Distribution Function</a>
 559                   </p>
 560                 </td>
 561 <td>
 562                   <p>
 563                     The probability of obtaining <span class="bold"><strong>more than
 564                     k successes</strong></span> from n trials with success fraction p.
 565                     For example:
 566                   </p>
 567                   <p>
 568                     <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>
 569                     <span class="identifier">p</span><span class="special">),</span>
 570                     <span class="identifier">k</span><span class="special">))</span></code>
 571                   </p>
 572                 </td>
 573 </tr>
 574 <tr>
 575 <td>
 576                   <p>
 577                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>
 578                   </p>
 579                 </td>
 580 <td>
 581                   <p>
 582                     The <span class="bold"><strong>greatest</strong></span> number of successes
 583                     that may be observed from n trials with success fraction p, at
 584                     probability P. Note that the value returned is a real-number,
 585                     and not an integer. Depending on the use case you may want to
 586                     take either the floor or ceiling of the result. For example:
 587                   </p>
 588                   <p>
 589                     <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>
 590                     <span class="identifier">p</span><span class="special">),</span>
 591                     <span class="identifier">P</span><span class="special">)</span></code>
 592                   </p>
 593                 </td>
 594 </tr>
 595 <tr>
 596 <td>
 597                   <p>
 598                     <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile
 599                     from the complement of the probability</a>
 600                   </p>
 601                 </td>
 602 <td>
 603                   <p>
 604                     The <span class="bold"><strong>smallest</strong></span> number of successes
 605                     that may be observed from n trials with success fraction p, at
 606                     probability P. Note that the value returned is a real-number,
 607                     and not an integer. Depending on the use case you may want to
 608                     take either the floor or ceiling of the result. For example:
 609                   </p>
 610                   <p>
 611                     <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>
 612                     <span class="identifier">p</span><span class="special">),</span>
 613                     <span class="identifier">P</span><span class="special">))</span></code>
 614                   </p>
 615                 </td>
 616 </tr>
 617 </tbody>
 618 </table></div>
 619 </div>
 620 <br class="table-break"><h5>
 621 <a name="math_toolkit.dist_ref.dists.binomial_dist.h8"></a>
 622           <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>
 623         </h5>
 624 <p>
 625           Various <a class="link" href="../../stat_tut/weg/binom_eg.html" title="Binomial Distribution Examples">worked examples</a>
 626           are available illustrating the use of the binomial distribution.
 627         </p>
 628 <h5>
 629 <a name="math_toolkit.dist_ref.dists.binomial_dist.h9"></a>
 630           <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>
 631         </h5>
 632 <p>
 633           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>,
 634           please refer to these functions for information on accuracy.
 635         </p>
 636 <h5>
 637 <a name="math_toolkit.dist_ref.dists.binomial_dist.h10"></a>
 638           <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>
 639         </h5>
 640 <p>
 641           In the following table <span class="emphasis"><em>p</em></span> is the probability that one
 642           trial will be successful (the success fraction), <span class="emphasis"><em>n</em></span>
 643           is the number of trials, <span class="emphasis"><em>k</em></span> is the number of successes,
 644           <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 645         </p>
 646 <div class="informaltable"><table class="table">
 647 <colgroup>
 648 <col>
 649 <col>
 650 </colgroup>
 651 <thead><tr>
 652 <th>
 653                   <p>
 654                     Function
 655                   </p>
 656                 </th>
 657 <th>
 658                   <p>
 659                     Implementation Notes
 660                   </p>
 661                 </th>
 662 </tr></thead>
 663 <tbody>
 664 <tr>
 665 <td>
 666                   <p>
 667                     pdf
 668                   </p>
 669                 </td>
 670 <td>
 671                   <p>
 672                     Implementation is in terms of <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>:
 673                     if <sub>n</sub>C<sub>k </sub> is the binomial coefficient of a and b, then we have:
 674                   </p>
 675                   <p>
 676                     <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref1.svg"></span>
 677                   </p>
 678                   <p>
 679                     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>
 680                     <span class="special">(</span><span class="identifier">n</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code>
 681                   </p>
 682                   <p>
 683                     The function <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>
 684                     is used here, since it has already been optimised for the lowest
 685                     possible error - indeed this is really just a thin wrapper around
 686                     part of the internals of the incomplete beta function.
 687                   </p>
 688                   <p>
 689                     There are also various special cases: refer to the code for details.
 690                   </p>
 691                 </td>
 692 </tr>
 693 <tr>
 694 <td>
 695                   <p>
 696                     cdf
 697                   </p>
 698                 </td>
 699 <td>
 700                   <p>
 701                     Using the relation:
 702                   </p>
 703 <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>
 704   <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>
 705   <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>
 706                   <p>
 707                     There are also various special cases: refer to the code for details.
 708                   </p>
 709                 </td>
 710 </tr>
 711 <tr>
 712 <td>
 713                   <p>
 714                     cdf complement
 715                   </p>
 716                 </td>
 717 <td>
 718                   <p>
 719                     Using the relation: q = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(k
 720                     + 1, n - k, p)
 721                   </p>
 722                   <p>
 723                     There are also various special cases: refer to the code for details.
 724                   </p>
 725                 </td>
 726 </tr>
 727 <tr>
 728 <td>
 729                   <p>
 730                     quantile
 731                   </p>
 732                 </td>
 733 <td>
 734                   <p>
 735                     Since the cdf is non-linear in variate <span class="emphasis"><em>k</em></span>
 736                     none of the inverse incomplete beta functions can be used here.
 737                     Instead the quantile is found numerically using a derivative
 738                     free method (<a class="link" href="../../roots/roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS
 739                     748 algorithm</a>).
 740                   </p>
 741                 </td>
 742 </tr>
 743 <tr>
 744 <td>
 745                   <p>
 746                     quantile from the complement
 747                   </p>
 748                 </td>
 749 <td>
 750                   <p>
 751                     Found numerically as above.
 752                   </p>
 753                 </td>
 754 </tr>
 755 <tr>
 756 <td>
 757                   <p>
 758                     mean
 759                   </p>
 760                 </td>
 761 <td>
 762                   <p>
 763                     <code class="computeroutput"><span class="identifier">p</span> <span class="special">*</span>
 764                     <span class="identifier">n</span></code>
 765                   </p>
 766                 </td>
 767 </tr>
 768 <tr>
 769 <td>
 770                   <p>
 771                     variance
 772                   </p>
 773                 </td>
 774 <td>
 775                   <p>
 776                     <code class="computeroutput"><span class="identifier">p</span> <span class="special">*</span>
 777                     <span class="identifier">n</span> <span class="special">*</span>
 778                     <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)</span></code>
 779                   </p>
 780                 </td>
 781 </tr>
 782 <tr>
 783 <td>
 784                   <p>
 785                     mode
 786                   </p>
 787                 </td>
 788 <td>
 789                   <p>
 790                     <code class="computeroutput"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">p</span> <span class="special">*</span>
 791                     <span class="special">(</span><span class="identifier">n</span>
 792                     <span class="special">+</span> <span class="number">1</span><span class="special">))</span></code>
 793                   </p>
 794                 </td>
 795 </tr>
 796 <tr>
 797 <td>
 798                   <p>
 799                     skewness
 800                   </p>
 801                 </td>
 802 <td>
 803                   <p>
 804                     <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
 805                     <span class="special">-</span> <span class="number">2</span>
 806                     <span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span>
 807                     <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">n</span> <span class="special">*</span>
 808                     <span class="identifier">p</span> <span class="special">*</span>
 809                     <span class="special">(</span><span class="number">1</span>
 810                     <span class="special">-</span> <span class="identifier">p</span><span class="special">))</span></code>
 811                   </p>
 812                 </td>
 813 </tr>
 814 <tr>
 815 <td>
 816                   <p>
 817                     kurtosis
 818                   </p>
 819                 </td>
 820 <td>
 821                   <p>
 822                     <code class="computeroutput"><span class="number">3</span> <span class="special">-</span>
 823                     <span class="special">(</span><span class="number">6</span>
 824                     <span class="special">/</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">+</span>
 825                     <span class="special">(</span><span class="number">1</span>
 826                     <span class="special">/</span> <span class="special">(</span><span class="identifier">n</span> <span class="special">*</span>
 827                     <span class="identifier">p</span> <span class="special">*</span>
 828                     <span class="special">(</span><span class="number">1</span>
 829                     <span class="special">-</span> <span class="identifier">p</span><span class="special">)))</span></code>
 830                   </p>
 831                 </td>
 832 </tr>
 833 <tr>
 834 <td>
 835                   <p>
 836                     kurtosis excess
 837                   </p>
 838                 </td>
 839 <td>
 840                   <p>
 841                     <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
 842                     <span class="special">-</span> <span class="number">6</span>
 843                     <span class="special">*</span> <span class="identifier">p</span>
 844                     <span class="special">*</span> <span class="identifier">q</span><span class="special">)</span> <span class="special">/</span>
 845                     <span class="special">(</span><span class="identifier">n</span>
 846                     <span class="special">*</span> <span class="identifier">p</span>
 847                     <span class="special">*</span> <span class="identifier">q</span><span class="special">)</span></code>
 848                   </p>
 849                 </td>
 850 </tr>
 851 <tr>
 852 <td>
 853                   <p>
 854                     parameter estimation
 855                   </p>
 856                 </td>
 857 <td>
 858                   <p>
 859                     The member functions <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
 860                     <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
 861                     and <code class="computeroutput"><span class="identifier">find_number_of_trials</span></code>
 862                     are implemented in terms of the inverse incomplete beta functions
 863                     <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>,
 864                     <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>,
 865                     and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_invb</a>
 866                     respectively
 867                   </p>
 868                 </td>
 869 </tr>
 870 </tbody>
 871 </table></div>
 872 <h5>
 873 <a name="math_toolkit.dist_ref.dists.binomial_dist.h11"></a>
 874           <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>
 875         </h5>
 876 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 877 <li class="listitem">
 878               <a href="http://mathworld.wolfram.com/BinomialDistribution.html" target="_top">Weisstein,
 879               Eric W. "Binomial Distribution." From MathWorld--A Wolfram
 880               Web Resource</a>.
 881             </li>
 882 <li class="listitem">
 883               <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia
 884               binomial distribution</a>.
 885             </li>
 886 <li class="listitem">
 887               <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm" target="_top">NIST
 888               Explorary Data Analysis</a>.
 889             </li>
 890 </ul></div>
 891 </div>
 892 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 893 <td align="left"></td>
 894 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 895       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 896       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 897       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 898         Distributed under the Boost Software License, Version 1.0. (See accompanying
 899         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>)
 900       </p>
 901 </div></td>
 902 </tr></table>
 903 <hr>
 904 <div class="spirit-nav">
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