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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>
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