1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Binomial Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="beta_dist.html" title="Beta Distribution"> 10<link rel="next" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="beta_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cauchy_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 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"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> 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"><></span> <span class="identifier">binomial</span><span class="special">;</span> 38 39<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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<div class="blockquote"><blockquote class="blockquote"><p> 108 <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref2.svg"></span> 109 110 </p></blockquote></div> 111<p> 112 The following two graphs illustrate how the PDF changes depending upon 113 the distributions parameters, first we'll keep the success fraction <span class="emphasis"><em>p</em></span> 114 fixed at 0.5, and vary the sample size: 115 </p> 116<div class="blockquote"><blockquote class="blockquote"><p> 117 <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_1.svg" align="middle"></span> 118 119 </p></blockquote></div> 120<p> 121 Alternatively, we can keep the sample size fixed at N=20 and vary the success 122 fraction <span class="emphasis"><em>p</em></span>: 123 </p> 124<div class="blockquote"><blockquote class="blockquote"><p> 125 <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_2.svg" align="middle"></span> 126 127 </p></blockquote></div> 128<div class="caution"><table border="0" summary="Caution"> 129<tr> 130<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> 131<th align="left">Caution</th> 132</tr> 133<tr><td align="left" valign="top"> 134<p> 135 The Binomial distribution is a discrete distribution: internally, functions 136 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 137 are continuous functions, but in reality the results returned from these 138 functions only have meaning if an integer value is provided for the random 139 variate argument. 140 </p> 141<p> 142 The quantile function will by default return an integer result that has 143 been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles 144 (where the probability is less than 0.5) are rounded downward, and upper 145 quantiles (where the probability is greater than 0.5) are rounded upwards. 146 This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at 147 least</em></span> the requested coverage will be present in the central 148 region, and <span class="emphasis"><em>no more than</em></span> the requested coverage 149 will be present in the tails. 150 </p> 151<p> 152 This behaviour can be changed so that the quantile functions are rounded 153 differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>. 154 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 155 Quantiles of Discrete Distributions</a> before using the quantile 156 function on the Binomial distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference 157 docs</a> describe how to change the rounding policy for these distributions. 158 </p> 159</td></tr> 160</table></div> 161<h5> 162<a name="math_toolkit.dist_ref.dists.binomial_dist.h0"></a> 163 <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 164 Functions</a> 165 </h5> 166<h6> 167<a name="math_toolkit.dist_ref.dists.binomial_dist.h1"></a> 168 <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> 169 </h6> 170<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> 171</pre> 172<p> 173 Constructor: <span class="emphasis"><em>n</em></span> is the total number of trials, <span class="emphasis"><em>p</em></span> 174 is the probability of success of a single trial. 175 </p> 176<p> 177 Requires <code class="computeroutput"><span class="number">0</span> <span class="special"><=</span> 178 <span class="identifier">p</span> <span class="special"><=</span> 179 <span class="number">1</span></code>, and <code class="computeroutput"><span class="identifier">n</span> 180 <span class="special">>=</span> <span class="number">0</span></code>, 181 otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 182 </p> 183<h6> 184<a name="math_toolkit.dist_ref.dists.binomial_dist.h2"></a> 185 <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> 186 </h6> 187<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> 188</pre> 189<p> 190 Returns the parameter <span class="emphasis"><em>p</em></span> from which this distribution 191 was constructed. 192 </p> 193<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> 194</pre> 195<p> 196 Returns the parameter <span class="emphasis"><em>n</em></span> from which this distribution 197 was constructed. 198 </p> 199<h6> 200<a name="math_toolkit.dist_ref.dists.binomial_dist.h3"></a> 201 <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 202 Bound on the Success Fraction</a> 203 </h6> 204<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> 205 <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> 206 <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> 207 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> 208 <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> 209</pre> 210<p> 211 Returns a lower bound on the success fraction: 212 </p> 213<div class="variablelist"> 214<p class="title"><b></b></p> 215<dl class="variablelist"> 216<dt><span class="term">trials</span></dt> 217<dd><p> 218 The total number of trials conducted. 219 </p></dd> 220<dt><span class="term">successes</span></dt> 221<dd><p> 222 The number of successes that occurred. 223 </p></dd> 224<dt><span class="term">alpha</span></dt> 225<dd><p> 226 The largest acceptable probability that the true value of the success 227 fraction is <span class="bold"><strong>less than</strong></span> the value 228 returned. 229 </p></dd> 230<dt><span class="term">method</span></dt> 231<dd><p> 232 An optional parameter that specifies the method to be used to compute 233 the interval (See below). 234 </p></dd> 235</dl> 236</div> 237<p> 238 For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span> 239 trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>, 240 but if you want to be 95% sure that the true value is <span class="bold"><strong>greater 241 than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then: 242 </p> 243<pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 244</pre> 245<p> 246 <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 247 example.</a> 248 </p> 249<p> 250 There are currently two possible values available for the <span class="emphasis"><em>method</em></span> 251 optional parameter: <span class="emphasis"><em>clopper_pearson_exact_interval</em></span> 252 or <span class="emphasis"><em>jeffreys_prior_interval</em></span>. These constants are both 253 members of class template <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>, 254 so usage is for example: 255 </p> 256<pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> 257 <span class="identifier">n</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span> 258</pre> 259<p> 260 The default method if this parameter is not specified is the Clopper Pearson 261 "exact" interval. This produces an interval that guarantees at 262 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 263 conservative, sometimes producing intervals with much greater than the 264 requested coverage. 265 </p> 266<p> 267 The alternative calculation method produces a non-informative Jeffreys 268 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> 269 coverage only <span class="emphasis"><em>in the average case</em></span>, though is typically 270 very close to the requested coverage level. It is one of the main methods 271 of calculation recommended in the review by Brown, Cai and DasGupta. 272 </p> 273<p> 274 Please note that the "textbook" calculation method using a normal 275 approximation (the Wald interval) is deliberately not provided: it is known 276 to produce consistently poor results, even when the sample size is surprisingly 277 large. Refer to Brown, Cai and DasGupta for a full explanation. Many other 278 methods of calculation are available, and may be more appropriate for specific 279 situations. Unfortunately there appears to be no consensus amongst statisticians 280 as to which is "best": refer to the discussion at the end of 281 Brown, Cai and DasGupta for examples. 282 </p> 283<p> 284 The two methods provided here were chosen principally because they can 285 be used for both one and two sided intervals. See also: 286 </p> 287<p> 288 Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001), Interval Estimation 289 for a Binomial Proportion, Statistical Science, Vol. 16, No. 2, 101-133. 290 </p> 291<p> 292 T. Tony Cai (2005), One-sided confidence intervals in discrete distributions, 293 Journal of Statistical Planning and Inference 131, 63-88. 294 </p> 295<p> 296 Agresti, A. and Coull, B. A. (1998). Approximate is better than "exact" 297 for interval estimation of binomial proportions. Amer. Statist. 52 119-126. 298 </p> 299<p> 300 Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial 301 limits illustrated in the case of the binomial. Biometrika 26 404-413. 302 </p> 303<h6> 304<a name="math_toolkit.dist_ref.dists.binomial_dist.h4"></a> 305 <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 306 Bound on the Success Fraction</a> 307 </h6> 308<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> 309 <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> 310 <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> 311 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> 312 <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> 313</pre> 314<p> 315 Returns an upper bound on the success fraction: 316 </p> 317<div class="variablelist"> 318<p class="title"><b></b></p> 319<dl class="variablelist"> 320<dt><span class="term">trials</span></dt> 321<dd><p> 322 The total number of trials conducted. 323 </p></dd> 324<dt><span class="term">successes</span></dt> 325<dd><p> 326 The number of successes that occurred. 327 </p></dd> 328<dt><span class="term">alpha</span></dt> 329<dd><p> 330 The largest acceptable probability that the true value of the success 331 fraction is <span class="bold"><strong>greater than</strong></span> the value 332 returned. 333 </p></dd> 334<dt><span class="term">method</span></dt> 335<dd><p> 336 An optional parameter that specifies the method to be used to compute 337 the interval. Refer to the documentation for <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code> 338 above for the meaning of the method options. 339 </p></dd> 340</dl> 341</div> 342<p> 343 For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span> 344 trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>, 345 but if you want to be 95% sure that the true value is <span class="bold"><strong>less 346 than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then: 347 </p> 348<pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 349</pre> 350<p> 351 <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 352 example.</a> 353 </p> 354<div class="note"><table border="0" summary="Note"> 355<tr> 356<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> 357<th align="left">Note</th> 358</tr> 359<tr><td align="left" valign="top"> 360<p> 361 In order to obtain a two sided bound on the success fraction, you call 362 both <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code> 363 <span class="bold"><strong>and</strong></span> <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code> 364 each with the same arguments. 365 </p> 366<p> 367 If the desired risk level that the true success fraction lies outside 368 the bounds is α, then you pass α/2 to these functions. 369 </p> 370<p> 371 So for example a two sided 95% confidence interval would be obtained 372 by passing α = 0.025 to each of the functions. 373 </p> 374<p> 375 <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 376 example.</a> 377 </p> 378</td></tr> 379</table></div> 380<h6> 381<a name="math_toolkit.dist_ref.dists.binomial_dist.h5"></a> 382 <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 383 the Number of Trials Required for a Certain Number of Successes</a> 384 </h6> 385<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> 386 <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of events</span> 387 <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction</span> 388 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold</span> 389</pre> 390<p> 391 This function estimates the minimum number of trials required to ensure 392 that more than k events is observed with a level of risk <span class="emphasis"><em>alpha</em></span> 393 that k or fewer events occur. 394 </p> 395<div class="variablelist"> 396<p class="title"><b></b></p> 397<dl class="variablelist"> 398<dt><span class="term">k</span></dt> 399<dd><p> 400 The number of success observed. 401 </p></dd> 402<dt><span class="term">p</span></dt> 403<dd><p> 404 The probability of success for each trial. 405 </p></dd> 406<dt><span class="term">alpha</span></dt> 407<dd><p> 408 The maximum acceptable probability that k events or fewer will be 409 observed. 410 </p></dd> 411</dl> 412</div> 413<p> 414 For example: 415 </p> 416<pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 417</pre> 418<p> 419 Returns the smallest number of trials we must conduct to be 95% sure of 420 seeing 10 events that occur with frequency one half. 421 </p> 422<h6> 423<a name="math_toolkit.dist_ref.dists.binomial_dist.h6"></a> 424 <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 425 the Maximum Number of Trials to Ensure no more than a Certain Number of 426 Successes</a> 427 </h6> 428<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> 429 <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of events</span> 430 <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction</span> 431 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold</span> 432</pre> 433<p> 434 This function estimates the maximum number of trials we can conduct to 435 ensure that k successes or fewer are observed, with a risk <span class="emphasis"><em>alpha</em></span> 436 that more than k occur. 437 </p> 438<div class="variablelist"> 439<p class="title"><b></b></p> 440<dl class="variablelist"> 441<dt><span class="term">k</span></dt> 442<dd><p> 443 The number of success observed. 444 </p></dd> 445<dt><span class="term">p</span></dt> 446<dd><p> 447 The probability of success for each trial. 448 </p></dd> 449<dt><span class="term">alpha</span></dt> 450<dd><p> 451 The maximum acceptable probability that more than k events will be 452 observed. 453 </p></dd> 454</dl> 455</div> 456<p> 457 For example: 458 </p> 459<pre class="programlisting"><span class="identifier">binomial_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1e-6</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 460</pre> 461<p> 462 Returns the largest number of trials we can conduct and still be 95% certain 463 of not observing any events that occur with one in a million frequency. 464 This is typically used in failure analysis. 465 </p> 466<p> 467 <a class="link" href="../../stat_tut/weg/binom_eg/binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution.">See Worked 468 Example.</a> 469 </p> 470<h5> 471<a name="math_toolkit.dist_ref.dists.binomial_dist.h7"></a> 472 <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 473 Accessors</a> 474 </h5> 475<p> 476 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 477 functions</a> that are generic to all distributions are supported: 478 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 479 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 480 <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>, 481 <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>, 482 <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>, 483 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 484 <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>, 485 <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>. 486 </p> 487<p> 488 The domain for the random variable <span class="emphasis"><em>k</em></span> is <code class="computeroutput"><span class="number">0</span> <span class="special"><=</span> <span class="identifier">k</span> <span class="special"><=</span> <span class="identifier">N</span></code>, otherwise a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 489 is returned. 490 </p> 491<p> 492 It's worth taking a moment to define what these accessors actually mean 493 in the context of this distribution: 494 </p> 495<div class="table"> 496<a name="math_toolkit.dist_ref.dists.binomial_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table 5.1. Meaning of the non-member accessors</b></p> 497<div class="table-contents"><table class="table" summary="Meaning of the non-member accessors"> 498<colgroup> 499<col> 500<col> 501</colgroup> 502<thead><tr> 503<th> 504 <p> 505 Function 506 </p> 507 </th> 508<th> 509 <p> 510 Meaning 511 </p> 512 </th> 513</tr></thead> 514<tbody> 515<tr> 516<td> 517 <p> 518 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density 519 Function</a> 520 </p> 521 </td> 522<td> 523 <p> 524 The probability of obtaining <span class="bold"><strong>exactly k 525 successes</strong></span> from n trials with success fraction p. For 526 example: 527 </p> 528 <p> 529 <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> 530 <span class="identifier">p</span><span class="special">),</span> 531 <span class="identifier">k</span><span class="special">)</span></code> 532 </p> 533 </td> 534</tr> 535<tr> 536<td> 537 <p> 538 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution 539 Function</a> 540 </p> 541 </td> 542<td> 543 <p> 544 The probability of obtaining <span class="bold"><strong>k successes 545 or fewer</strong></span> from n trials with success fraction p. For 546 example: 547 </p> 548 <p> 549 <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> 550 <span class="identifier">p</span><span class="special">),</span> 551 <span class="identifier">k</span><span class="special">)</span></code> 552 </p> 553 </td> 554</tr> 555<tr> 556<td> 557 <p> 558 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of 559 the Cumulative Distribution Function</a> 560 </p> 561 </td> 562<td> 563 <p> 564 The probability of obtaining <span class="bold"><strong>more than 565 k successes</strong></span> from n trials with success fraction p. 566 For example: 567 </p> 568 <p> 569 <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> 570 <span class="identifier">p</span><span class="special">),</span> 571 <span class="identifier">k</span><span class="special">))</span></code> 572 </p> 573 </td> 574</tr> 575<tr> 576<td> 577 <p> 578 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a> 579 </p> 580 </td> 581<td> 582 <p> 583 Given a binomial distribution with <span class="emphasis"><em>n</em></span> trials, 584 success fraction <span class="emphasis"><em>p</em></span> and probability <span class="emphasis"><em>P</em></span>, 585 finds the largest number of successes <span class="emphasis"><em>k</em></span> 586 whose CDF is less than <span class="emphasis"><em>P</em></span>. It is strongly 587 recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding 588 Quantiles of Discrete Distributions</a> before using the quantile 589 function. 590 </p> 591 </td> 592</tr> 593<tr> 594<td> 595 <p> 596 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile 597 from the complement of the probability</a> 598 </p> 599 </td> 600<td> 601 <p> 602 Given a binomial distribution with <span class="emphasis"><em>n</em></span> trials, 603 success fraction <span class="emphasis"><em>p</em></span> and probability <span class="emphasis"><em>Q</em></span>, 604 finds the smallest number of successes <span class="emphasis"><em>k</em></span> 605 whose CDF is greater than <span class="emphasis"><em>1-Q</em></span>. It is strongly 606 recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding 607 Quantiles of Discrete Distributions</a> before using the quantile 608 function. 609 </p> 610 </td> 611</tr> 612</tbody> 613</table></div> 614</div> 615<br class="table-break"><h5> 616<a name="math_toolkit.dist_ref.dists.binomial_dist.h8"></a> 617 <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> 618 </h5> 619<p> 620 Various <a class="link" href="../../stat_tut/weg/binom_eg.html" title="Binomial Distribution Examples">worked examples</a> 621 are available illustrating the use of the binomial distribution. 622 </p> 623<h5> 624<a name="math_toolkit.dist_ref.dists.binomial_dist.h9"></a> 625 <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> 626 </h5> 627<p> 628 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>, 629 please refer to these functions for information on accuracy. 630 </p> 631<h5> 632<a name="math_toolkit.dist_ref.dists.binomial_dist.h10"></a> 633 <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> 634 </h5> 635<p> 636 In the following table <span class="emphasis"><em>p</em></span> is the probability that one 637 trial will be successful (the success fraction), <span class="emphasis"><em>n</em></span> 638 is the number of trials, <span class="emphasis"><em>k</em></span> is the number of successes, 639 <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>. 640 </p> 641<div class="informaltable"><table class="table"> 642<colgroup> 643<col> 644<col> 645</colgroup> 646<thead><tr> 647<th> 648 <p> 649 Function 650 </p> 651 </th> 652<th> 653 <p> 654 Implementation Notes 655 </p> 656 </th> 657</tr></thead> 658<tbody> 659<tr> 660<td> 661 <p> 662 pdf 663 </p> 664 </td> 665<td> 666 <p> 667 Implementation is in terms of <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>: 668 if <sub>n</sub>C<sub>k </sub> is the binomial coefficient of a and b, then we have: 669 </p> 670 <div class="blockquote"><blockquote class="blockquote"><p> 671 <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref1.svg"></span> 672 673 </p></blockquote></div> 674 <p> 675 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> 676 <span class="special">(</span><span class="identifier">n</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code> 677 </p> 678 <p> 679 The function <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a> 680 is used here, since it has already been optimised for the lowest 681 possible error - indeed this is really just a thin wrapper around 682 part of the internals of the incomplete beta function. 683 </p> 684 <p> 685 There are also various special cases: refer to the code for details. 686 </p> 687 </td> 688</tr> 689<tr> 690<td> 691 <p> 692 cdf 693 </p> 694 </td> 695<td> 696 <p> 697 Using the relation: 698 </p> 699<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> 700 <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> 701 <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> 702 <p> 703 There are also various special cases: refer to the code for details. 704 </p> 705 </td> 706</tr> 707<tr> 708<td> 709 <p> 710 cdf complement 711 </p> 712 </td> 713<td> 714 <p> 715 Using the relation: q = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(k 716 + 1, n - k, p) 717 </p> 718 <p> 719 There are also various special cases: refer to the code for details. 720 </p> 721 </td> 722</tr> 723<tr> 724<td> 725 <p> 726 quantile 727 </p> 728 </td> 729<td> 730 <p> 731 Since the cdf is non-linear in variate <span class="emphasis"><em>k</em></span> 732 none of the inverse incomplete beta functions can be used here. 733 Instead the quantile is found numerically using a derivative 734 free method (<a class="link" href="../../roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 735 748 algorithm</a>). 736 </p> 737 </td> 738</tr> 739<tr> 740<td> 741 <p> 742 quantile from the complement 743 </p> 744 </td> 745<td> 746 <p> 747 Found numerically as above. 748 </p> 749 </td> 750</tr> 751<tr> 752<td> 753 <p> 754 mean 755 </p> 756 </td> 757<td> 758 <p> 759 <code class="computeroutput"><span class="identifier">p</span> <span class="special">*</span> 760 <span class="identifier">n</span></code> 761 </p> 762 </td> 763</tr> 764<tr> 765<td> 766 <p> 767 variance 768 </p> 769 </td> 770<td> 771 <p> 772 <code class="computeroutput"><span class="identifier">p</span> <span class="special">*</span> 773 <span class="identifier">n</span> <span class="special">*</span> 774 <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">)</span></code> 775 </p> 776 </td> 777</tr> 778<tr> 779<td> 780 <p> 781 mode 782 </p> 783 </td> 784<td> 785 <p> 786 <code class="computeroutput"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">p</span> <span class="special">*</span> 787 <span class="special">(</span><span class="identifier">n</span> 788 <span class="special">+</span> <span class="number">1</span><span class="special">))</span></code> 789 </p> 790 </td> 791</tr> 792<tr> 793<td> 794 <p> 795 skewness 796 </p> 797 </td> 798<td> 799 <p> 800 <code class="computeroutput"><span class="special">(</span><span class="number">1</span> 801 <span class="special">-</span> <span class="number">2</span> 802 <span class="special">*</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span> 803 <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">n</span> <span class="special">*</span> 804 <span class="identifier">p</span> <span class="special">*</span> 805 <span class="special">(</span><span class="number">1</span> 806 <span class="special">-</span> <span class="identifier">p</span><span class="special">))</span></code> 807 </p> 808 </td> 809</tr> 810<tr> 811<td> 812 <p> 813 kurtosis 814 </p> 815 </td> 816<td> 817 <p> 818 <code class="computeroutput"><span class="number">3</span> <span class="special">-</span> 819 <span class="special">(</span><span class="number">6</span> 820 <span class="special">/</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">+</span> 821 <span class="special">(</span><span class="number">1</span> 822 <span class="special">/</span> <span class="special">(</span><span class="identifier">n</span> <span class="special">*</span> 823 <span class="identifier">p</span> <span class="special">*</span> 824 <span class="special">(</span><span class="number">1</span> 825 <span class="special">-</span> <span class="identifier">p</span><span class="special">)))</span></code> 826 </p> 827 </td> 828</tr> 829<tr> 830<td> 831 <p> 832 kurtosis excess 833 </p> 834 </td> 835<td> 836 <p> 837 <code class="computeroutput"><span class="special">(</span><span class="number">1</span> 838 <span class="special">-</span> <span class="number">6</span> 839 <span class="special">*</span> <span class="identifier">p</span> 840 <span class="special">*</span> <span class="identifier">q</span><span class="special">)</span> <span class="special">/</span> 841 <span class="special">(</span><span class="identifier">n</span> 842 <span class="special">*</span> <span class="identifier">p</span> 843 <span class="special">*</span> <span class="identifier">q</span><span class="special">)</span></code> 844 </p> 845 </td> 846</tr> 847<tr> 848<td> 849 <p> 850 parameter estimation 851 </p> 852 </td> 853<td> 854 <p> 855 The member functions <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code> 856 <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code> 857 and <code class="computeroutput"><span class="identifier">find_number_of_trials</span></code> 858 are implemented in terms of the inverse incomplete beta functions 859 <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>, 860 <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>, 861 and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_invb</a> 862 respectively 863 </p> 864 </td> 865</tr> 866</tbody> 867</table></div> 868<h5> 869<a name="math_toolkit.dist_ref.dists.binomial_dist.h11"></a> 870 <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> 871 </h5> 872<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 873<li class="listitem"> 874 <a href="http://mathworld.wolfram.com/BinomialDistribution.html" target="_top">Weisstein, 875 Eric W. "Binomial Distribution." From MathWorld--A Wolfram 876 Web Resource</a>. 877 </li> 878<li class="listitem"> 879 <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia 880 binomial distribution</a>. 881 </li> 882<li class="listitem"> 883 <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm" target="_top">NIST 884 Exploratory Data Analysis</a>. 885 </li> 886</ul></div> 887</div> 888<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 889<td align="left"></td> 890<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 891 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 892 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 893 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 894 Daryle Walker and Xiaogang Zhang<p> 895 Distributed under the Boost Software License, Version 1.0. (See accompanying 896 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>) 897 </p> 898</div></td> 899</tr></table> 900<hr> 901<div class="spirit-nav"> 902<a accesskey="p" href="beta_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cauchy_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 903</div> 904</body> 905</html> 906
