1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Geometric 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="gamma_dist.html" title="Gamma (and Erlang) Distribution"> 10<link rel="next" href="hyperexponential_dist.html" title="Hyperexponential 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="gamma_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="hyperexponential_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.geometric_dist"></a><a class="link" href="geometric_dist.html" title="Geometric Distribution">Geometric 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">geometric</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">geometric_distribution</span><span class="special">;</span> 36 37<span class="keyword">typedef</span> <span class="identifier">geometric_distribution</span><span class="special"><></span> <span class="identifier">geometric</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">geometric_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 <span class="comment">// Constructor from success_fraction:</span> 46 <span class="identifier">geometric_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> 47 48 <span class="comment">// Parameter accessors:</span> 49 <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> 50 <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> 51 52 <span class="comment">// Bounds on success fraction:</span> 53 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> 54 <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> 55 <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> 56 <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha</span> 57 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_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> <span class="comment">// alpha</span> 61 62 <span class="comment">// Estimate min/max number of trials:</span> 63 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span> 64 <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// Number of failures.</span> 65 <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// Success fraction.</span> 66 <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha.</span> 67 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> 68 <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// Number of failures.</span> 69 <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// Success fraction.</span> 70 <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha.</span> 71<span class="special">};</span> 72 73<span class="special">}}</span> <span class="comment">// namespaces</span> 74</pre> 75<p> 76 The class type <code class="computeroutput"><span class="identifier">geometric_distribution</span></code> 77 represents a <a href="http://en.wikipedia.org/wiki/geometric_distribution" target="_top">geometric 78 distribution</a>: it is used when there are exactly two mutually exclusive 79 outcomes of a <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli 80 trial</a>: these outcomes are labelled "success" and "failure". 81 </p> 82<p> 83 For <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli 84 trials</a> each with success fraction <span class="emphasis"><em>p</em></span>, the geometric 85 distribution gives the probability of observing <span class="emphasis"><em>k</em></span> 86 trials (failures, events, occurrences, or arrivals) before the first success. 87 </p> 88<div class="note"><table border="0" summary="Note"> 89<tr> 90<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> 91<th align="left">Note</th> 92</tr> 93<tr><td align="left" valign="top"><p> 94 For this implementation, the set of trials <span class="bold"><strong>includes 95 zero</strong></span> (unlike another definition where the set of trials starts 96 at one, sometimes named <span class="emphasis"><em>shifted</em></span>). 97 </p></td></tr> 98</table></div> 99<p> 100 The geometric distribution assumes that success_fraction <span class="emphasis"><em>p</em></span> 101 is fixed for all <span class="emphasis"><em>k</em></span> trials. 102 </p> 103<p> 104 The probability that there are <span class="emphasis"><em>k</em></span> failures before the 105 first success 106 </p> 107<div class="blockquote"><blockquote class="blockquote"><p> 108 <span class="serif_italic">Pr(Y=<span class="emphasis"><em>k</em></span>) = (1-<span class="emphasis"><em>p</em></span>)<sup><span class="emphasis"><em>k</em></span></sup> <span class="emphasis"><em>p</em></span></span> 109 </p></blockquote></div> 110<p> 111 For example, when throwing a 6-face dice the success probability <span class="emphasis"><em>p</em></span> 112 = 1/6 = 0.1666 ̇. Throwing repeatedly until a <span class="emphasis"><em>three</em></span> 113 appears, the probability distribution of the number of times <span class="emphasis"><em>not-a-three</em></span> 114 is thrown is geometric. 115 </p> 116<p> 117 Geometric distribution has the Probability Density Function PDF: 118 </p> 119<div class="blockquote"><blockquote class="blockquote"><p> 120 <span class="serif_italic">(1-<span class="emphasis"><em>p</em></span>)<sup><span class="emphasis"><em>k</em></span></sup> <span class="emphasis"><em>p</em></span></span> 121 </p></blockquote></div> 122<p> 123 The following graph illustrates how the PDF and CDF vary for three examples 124 of the success fraction <span class="emphasis"><em>p</em></span>, (when considering the geometric 125 distribution as a continuous function), 126 </p> 127<div class="blockquote"><blockquote class="blockquote"><p> 128 <span class="inlinemediaobject"><img src="../../../../graphs/geometric_pdf_2.svg" align="middle"></span> 129 130 </p></blockquote></div> 131<div class="blockquote"><blockquote class="blockquote"><p> 132 <span class="inlinemediaobject"><img src="../../../../graphs/geometric_cdf_2.svg" align="middle"></span> 133 134 </p></blockquote></div> 135<p> 136 and as discrete. 137 </p> 138<div class="blockquote"><blockquote class="blockquote"><p> 139 <span class="inlinemediaobject"><img src="../../../../graphs/geometric_pdf_discrete.svg" align="middle"></span> 140 141 </p></blockquote></div> 142<div class="blockquote"><blockquote class="blockquote"><p> 143 <span class="inlinemediaobject"><img src="../../../../graphs/geometric_cdf_discrete.svg" align="middle"></span> 144 145 </p></blockquote></div> 146<h5> 147<a name="math_toolkit.dist_ref.dists.geometric_dist.h0"></a> 148 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.related_distributions"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.related_distributions">Related 149 Distributions</a> 150 </h5> 151<p> 152 The geometric distribution is a special case of the <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 153 Binomial Distribution</a> with successes parameter <span class="emphasis"><em>r</em></span> 154 = 1, so only one first and only success is required : thus by definition 155 <code class="computeroutput"><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special">==</span> 156 <span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span></code> 157 </p> 158<pre class="programlisting"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">);</span> 159<span class="identifier">negative_binomial</span> <span class="identifier">nb</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> 160<span class="identifier">geometric</span> <span class="identifier">g</span><span class="special">(</span><span class="identifier">success_fraction</span><span class="special">);</span> 161<span class="identifier">ASSERT</span><span class="special">(</span><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="number">1</span><span class="special">));</span> 162</pre> 163<p> 164 This implementation uses real numbers for the computation throughout (because 165 it uses the <span class="bold"><strong>real-valued</strong></span> power and exponential 166 functions). So to obtain a conventional strictly-discrete geometric distribution 167 you must ensure that an integer value is provided for the number of trials 168 (random variable) <span class="emphasis"><em>k</em></span>, and take integer values (floor 169 or ceil functions) from functions that return a number of successes. 170 </p> 171<div class="caution"><table border="0" summary="Caution"> 172<tr> 173<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> 174<th align="left">Caution</th> 175</tr> 176<tr><td align="left" valign="top"> 177<p> 178 The geometric distribution is a discrete distribution: internally, functions 179 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 180 are continuous functions, but in reality the results returned from these 181 functions only have meaning if an integer value is provided for the random 182 variate argument. 183 </p> 184<p> 185 The quantile function will by default return an integer result that has 186 been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles 187 (where the probability is less than 0.5) are rounded downward, and upper 188 quantiles (where the probability is greater than 0.5) are rounded upwards. 189 This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at 190 least</em></span> the requested coverage will be present in the central 191 region, and <span class="emphasis"><em>no more than</em></span> the requested coverage 192 will be present in the tails. 193 </p> 194<p> 195 This behaviour can be changed so that the quantile functions are rounded 196 differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>. 197 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 198 Quantiles of Discrete Distributions</a> before using the quantile 199 function on the geometric distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference 200 docs</a> describe how to change the rounding policy for these distributions. 201 </p> 202</td></tr> 203</table></div> 204<h5> 205<a name="math_toolkit.dist_ref.dists.geometric_dist.h1"></a> 206 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.member_functions"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.member_functions">Member 207 Functions</a> 208 </h5> 209<h6> 210<a name="math_toolkit.dist_ref.dists.geometric_dist.h2"></a> 211 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.constructor"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.constructor">Constructor</a> 212 </h6> 213<pre class="programlisting"><span class="identifier">geometric_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> 214</pre> 215<p> 216 Constructor: <span class="emphasis"><em>p</em></span> or success_fraction is the probability 217 of success of a single trial. 218 </p> 219<p> 220 Requires: <code class="computeroutput"><span class="number">0</span> <span class="special"><=</span> 221 <span class="identifier">p</span> <span class="special"><=</span> 222 <span class="number">1</span></code>. 223 </p> 224<h6> 225<a name="math_toolkit.dist_ref.dists.geometric_dist.h3"></a> 226 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.accessors"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.accessors">Accessors</a> 227 </h6> 228<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> <span class="comment">// successes / trials (0 <= p <= 1)</span> 229</pre> 230<p> 231 Returns the success_fraction parameter <span class="emphasis"><em>p</em></span> from which 232 this distribution was constructed. 233 </p> 234<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// required successes always one,</span> 235<span class="comment">// included for compatibility with negative binomial distribution</span> 236<span class="comment">// with successes r == 1.</span> 237</pre> 238<p> 239 Returns unity. 240 </p> 241<p> 242 The following functions are equivalent to those provided for the negative 243 binomial, with successes = 1, but are provided here for completeness. 244 </p> 245<p> 246 The best method of calculation for the following functions is disputed: 247 see <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial 248 Distribution</a> and <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 249 Binomial Distribution</a> for more discussion. 250 </p> 251<h6> 252<a name="math_toolkit.dist_ref.dists.geometric_dist.h4"></a> 253 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.lower_bound_on_success_fraction_"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.lower_bound_on_success_fraction_">Lower 254 Bound on success_fraction Parameter <span class="emphasis"><em>p</em></span></a> 255 </h6> 256<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> 257 <span class="identifier">RealType</span> <span class="identifier">failures</span><span class="special">,</span> 258 <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">)</span> <span class="comment">// (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.</span> 259</pre> 260<p> 261 Returns a <span class="bold"><strong>lower bound</strong></span> on the success fraction: 262 </p> 263<div class="variablelist"> 264<p class="title"><b></b></p> 265<dl class="variablelist"> 266<dt><span class="term">failures</span></dt> 267<dd><p> 268 The total number of failures before the 1st success. 269 </p></dd> 270<dt><span class="term">alpha</span></dt> 271<dd><p> 272 The largest acceptable probability that the true value of the success 273 fraction is <span class="bold"><strong>less than</strong></span> the value 274 returned. 275 </p></dd> 276</dl> 277</div> 278<p> 279 For example, if you observe <span class="emphasis"><em>k</em></span> failures from <span class="emphasis"><em>n</em></span> 280 trials the best estimate for the success fraction is simply 1/<span class="emphasis"><em>n</em></span>, 281 but if you want to be 95% sure that the true value is <span class="bold"><strong>greater 282 than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then: 283 </p> 284<pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">geometric_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span> 285 <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 286</pre> 287<p> 288 <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See 289 negative_binomial confidence interval example.</a> 290 </p> 291<p> 292 This function uses the Clopper-Pearson method of computing the lower bound 293 on the success fraction, whilst many texts refer to this method as giving 294 an "exact" result in practice it produces an interval that guarantees 295 <span class="emphasis"><em>at least</em></span> the coverage required, and may produce pessimistic 296 estimates for some combinations of <span class="emphasis"><em>failures</em></span> and <span class="emphasis"><em>successes</em></span>. 297 See: 298 </p> 299<p> 300 <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong 301 Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some 302 Discrete Distributions. Computational statistics and data analysis, 2005, 303 vol. 48, no3, 605-621</a>. 304 </p> 305<h6> 306<a name="math_toolkit.dist_ref.dists.geometric_dist.h5"></a> 307 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.upper_bound_on_success_fraction_"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.upper_bound_on_success_fraction_">Upper 308 Bound on success_fraction Parameter p</a> 309 </h6> 310<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> 311 <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> 312 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.</span> 313</pre> 314<p> 315 Returns an <span class="bold"><strong>upper bound</strong></span> on the success 316 fraction: 317 </p> 318<div class="variablelist"> 319<p class="title"><b></b></p> 320<dl class="variablelist"> 321<dt><span class="term">trials</span></dt> 322<dd><p> 323 The total number of trials conducted. 324 </p></dd> 325<dt><span class="term">alpha</span></dt> 326<dd><p> 327 The largest acceptable probability that the true value of the success 328 fraction is <span class="bold"><strong>greater than</strong></span> the value 329 returned. 330 </p></dd> 331</dl> 332</div> 333<p> 334 For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span> 335 trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>, 336 but if you want to be 95% sure that the true value is <span class="bold"><strong>less 337 than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then: 338 </p> 339<pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">geometric_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> 340 <span class="identifier">k</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 341</pre> 342<p> 343 <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See 344 negative binomial confidence interval example.</a> 345 </p> 346<p> 347 This function uses the Clopper-Pearson method of computing the lower bound 348 on the success fraction, whilst many texts refer to this method as giving 349 an "exact" result in practice it produces an interval that guarantees 350 <span class="emphasis"><em>at least</em></span> the coverage required, and may produce pessimistic 351 estimates for some combinations of <span class="emphasis"><em>failures</em></span> and <span class="emphasis"><em>successes</em></span>. 352 See: 353 </p> 354<p> 355 <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong 356 Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some 357 Discrete Distributions. Computational statistics and data analysis, 2005, 358 vol. 48, no3, 605-621</a>. 359 </p> 360<h6> 361<a name="math_toolkit.dist_ref.dists.geometric_dist.h6"></a> 362 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_e"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_e">Estimating 363 Number of Trials to Ensure at Least a Certain Number of Failures</a> 364 </h6> 365<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> 366 <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of failures.</span> 367 <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction.</span> 368 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%).</span> 369</pre> 370<p> 371 This functions estimates the number of trials required to achieve a certain 372 probability that <span class="bold"><strong>more than <span class="emphasis"><em>k</em></span> 373 failures will be observed</strong></span>. 374 </p> 375<div class="variablelist"> 376<p class="title"><b></b></p> 377<dl class="variablelist"> 378<dt><span class="term">k</span></dt> 379<dd><p> 380 The target number of failures to be observed. 381 </p></dd> 382<dt><span class="term">p</span></dt> 383<dd><p> 384 The probability of <span class="emphasis"><em>success</em></span> for each trial. 385 </p></dd> 386<dt><span class="term">alpha</span></dt> 387<dd><p> 388 The maximum acceptable <span class="emphasis"><em>risk</em></span> that only <span class="emphasis"><em>k</em></span> 389 failures or fewer will be observed. 390 </p></dd> 391</dl> 392</div> 393<p> 394 For example: 395 </p> 396<pre class="programlisting"><span class="identifier">geometric_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_minimum_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> 397</pre> 398<p> 399 Returns the smallest number of trials we must conduct to be 95% (1-0.05) 400 sure of seeing 10 failures that occur with frequency one half. 401 </p> 402<p> 403 <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial.">Worked 404 Example.</a> 405 </p> 406<p> 407 This function uses numeric inversion of the geometric distribution to obtain 408 the result: another interpretation of the result is that it finds the number 409 of trials (failures) that will lead to an <span class="emphasis"><em>alpha</em></span> probability 410 of observing <span class="emphasis"><em>k</em></span> failures or fewer. 411 </p> 412<h6> 413<a name="math_toolkit.dist_ref.dists.geometric_dist.h7"></a> 414 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_0"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_0">Estimating 415 Number of Trials to Ensure a Maximum Number of Failures or Less</a> 416 </h6> 417<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> 418 <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of failures.</span> 419 <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction.</span> 420 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%).</span> 421</pre> 422<p> 423 This functions estimates the maximum number of trials we can conduct and 424 achieve a certain probability that <span class="bold"><strong>k failures or 425 fewer will be observed</strong></span>. 426 </p> 427<div class="variablelist"> 428<p class="title"><b></b></p> 429<dl class="variablelist"> 430<dt><span class="term">k</span></dt> 431<dd><p> 432 The maximum number of failures to be observed. 433 </p></dd> 434<dt><span class="term">p</span></dt> 435<dd><p> 436 The probability of <span class="emphasis"><em>success</em></span> for each trial. 437 </p></dd> 438<dt><span class="term">alpha</span></dt> 439<dd><p> 440 The maximum acceptable <span class="emphasis"><em>risk</em></span> that more than 441 <span class="emphasis"><em>k</em></span> failures will be observed. 442 </p></dd> 443</dl> 444</div> 445<p> 446 For example: 447 </p> 448<pre class="programlisting"><span class="identifier">geometric_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">1.0</span><span class="special">-</span><span class="number">1.0</span><span class="special">/</span><span class="number">1000000</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> 449</pre> 450<p> 451 Returns the largest number of trials we can conduct and still be 95% sure 452 of seeing no failures that occur with frequency one in one million. 453 </p> 454<p> 455 This function uses numeric inversion of the geometric distribution to obtain 456 the result: another interpretation of the result, is that it finds the 457 number of trials that will lead to an <span class="emphasis"><em>alpha</em></span> probability 458 of observing more than k failures. 459 </p> 460<h5> 461<a name="math_toolkit.dist_ref.dists.geometric_dist.h8"></a> 462 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.non_member_accessors"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.non_member_accessors">Non-member 463 Accessors</a> 464 </h5> 465<p> 466 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 467 functions</a> that are generic to all distributions are supported: 468 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 469 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 470 <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>, 471 <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>, 472 <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>, 473 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 474 <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>, 475 <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>. 476 </p> 477<p> 478 However it's worth taking a moment to define what these actually mean in 479 the context of this distribution: 480 </p> 481<div class="table"> 482<a name="math_toolkit.dist_ref.dists.geometric_dist.meaning_of_the_non_member_access"></a><p class="title"><b>Table 5.2. Meaning of the non-member accessors.</b></p> 483<div class="table-contents"><table class="table" summary="Meaning of the non-member accessors."> 484<colgroup> 485<col> 486<col> 487</colgroup> 488<thead><tr> 489<th> 490 <p> 491 Function 492 </p> 493 </th> 494<th> 495 <p> 496 Meaning 497 </p> 498 </th> 499</tr></thead> 500<tbody> 501<tr> 502<td> 503 <p> 504 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density 505 Function</a> 506 </p> 507 </td> 508<td> 509 <p> 510 The probability of obtaining <span class="bold"><strong>exactly k 511 failures</strong></span> from <span class="emphasis"><em>k</em></span> trials with success 512 fraction p. For example: 513 </p> 514<pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre> 515 </td> 516</tr> 517<tr> 518<td> 519 <p> 520 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution 521 Function</a> 522 </p> 523 </td> 524<td> 525 <p> 526 The probability of obtaining <span class="bold"><strong>k failures 527 or fewer</strong></span> from <span class="emphasis"><em>k</em></span> trials with success 528 fraction p and success on the last trial. For example: 529 </p> 530<pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre> 531 </td> 532</tr> 533<tr> 534<td> 535 <p> 536 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of 537 the Cumulative Distribution Function</a> 538 </p> 539 </td> 540<td> 541 <p> 542 The probability of obtaining <span class="bold"><strong>more than 543 k failures</strong></span> from <span class="emphasis"><em>k</em></span> trials with 544 success fraction p and success on the last trial. For example: 545 </p> 546<pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">))</span></pre> 547 </td> 548</tr> 549<tr> 550<td> 551 <p> 552 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a> 553 </p> 554 </td> 555<td> 556 <p> 557 The <span class="bold"><strong>greatest</strong></span> number of failures 558 <span class="emphasis"><em>k</em></span> expected to be observed from <span class="emphasis"><em>k</em></span> 559 trials with success fraction <span class="emphasis"><em>p</em></span>, at probability 560 <span class="emphasis"><em>P</em></span>. Note that the value returned is a real-number, 561 and not an integer. Depending on the use case you may want to 562 take either the floor or ceiling of the real result. For example: 563 </p> 564<pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">)</span></pre> 565 </td> 566</tr> 567<tr> 568<td> 569 <p> 570 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile 571 from the complement of the probability</a> 572 </p> 573 </td> 574<td> 575 <p> 576 The <span class="bold"><strong>smallest</strong></span> number of failures 577 <span class="emphasis"><em>k</em></span> expected to be observed from <span class="emphasis"><em>k</em></span> 578 trials with success fraction <span class="emphasis"><em>p</em></span>, at probability 579 <span class="emphasis"><em>P</em></span>. Note that the value returned is a real-number, 580 and not an integer. Depending on the use case you may want to 581 take either the floor or ceiling of the real result. For example: 582 </p> 583<pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">))</span></pre> 584 </td> 585</tr> 586</tbody> 587</table></div> 588</div> 589<br class="table-break"><h5> 590<a name="math_toolkit.dist_ref.dists.geometric_dist.h9"></a> 591 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.accuracy"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.accuracy">Accuracy</a> 592 </h5> 593<p> 594 This distribution is implemented using the pow and exp functions, so most 595 results are accurate within a few epsilon for the RealType. For extreme 596 values of <code class="computeroutput"><span class="keyword">double</span></code> <span class="emphasis"><em>p</em></span>, 597 for example 0.9999999999, accuracy can fall significantly, for example 598 to 10 decimal digits (from 16). 599 </p> 600<h5> 601<a name="math_toolkit.dist_ref.dists.geometric_dist.h10"></a> 602 <span class="phrase"><a name="math_toolkit.dist_ref.dists.geometric_dist.implementation"></a></span><a class="link" href="geometric_dist.html#math_toolkit.dist_ref.dists.geometric_dist.implementation">Implementation</a> 603 </h5> 604<p> 605 In the following table, <span class="emphasis"><em>p</em></span> is the probability that 606 any one trial will be successful (the success fraction), <span class="emphasis"><em>k</em></span> 607 is the number of failures, <span class="emphasis"><em>p</em></span> is the probability and 608 <span class="emphasis"><em>q = 1-p</em></span>, <span class="emphasis"><em>x</em></span> is the given probability 609 to estimate the expected number of failures using the quantile. 610 </p> 611<div class="informaltable"><table class="table"> 612<colgroup> 613<col> 614<col> 615</colgroup> 616<thead><tr> 617<th> 618 <p> 619 Function 620 </p> 621 </th> 622<th> 623 <p> 624 Implementation Notes 625 </p> 626 </th> 627</tr></thead> 628<tbody> 629<tr> 630<td> 631 <p> 632 pdf 633 </p> 634 </td> 635<td> 636 <p> 637 pdf = p * pow(q, k) 638 </p> 639 </td> 640</tr> 641<tr> 642<td> 643 <p> 644 cdf 645 </p> 646 </td> 647<td> 648 <p> 649 cdf = 1 - q<sup>k=1</sup> 650 </p> 651 </td> 652</tr> 653<tr> 654<td> 655 <p> 656 cdf complement 657 </p> 658 </td> 659<td> 660 <p> 661 exp(log1p(-p) * (k+1)) 662 </p> 663 </td> 664</tr> 665<tr> 666<td> 667 <p> 668 quantile 669 </p> 670 </td> 671<td> 672 <p> 673 k = log1p(-x) / log1p(-p) -1 674 </p> 675 </td> 676</tr> 677<tr> 678<td> 679 <p> 680 quantile from the complement 681 </p> 682 </td> 683<td> 684 <p> 685 k = log(x) / log1p(-p) -1 686 </p> 687 </td> 688</tr> 689<tr> 690<td> 691 <p> 692 mean 693 </p> 694 </td> 695<td> 696 <p> 697 (1-p)/p 698 </p> 699 </td> 700</tr> 701<tr> 702<td> 703 <p> 704 variance 705 </p> 706 </td> 707<td> 708 <p> 709 (1-p)/p² 710 </p> 711 </td> 712</tr> 713<tr> 714<td> 715 <p> 716 mode 717 </p> 718 </td> 719<td> 720 <p> 721 0 722 </p> 723 </td> 724</tr> 725<tr> 726<td> 727 <p> 728 skewness 729 </p> 730 </td> 731<td> 732 <p> 733 (2-p)/√q 734 </p> 735 </td> 736</tr> 737<tr> 738<td> 739 <p> 740 kurtosis 741 </p> 742 </td> 743<td> 744 <p> 745 9+p²/q 746 </p> 747 </td> 748</tr> 749<tr> 750<td> 751 <p> 752 kurtosis excess 753 </p> 754 </td> 755<td> 756 <p> 757 6 +p²/q 758 </p> 759 </td> 760</tr> 761<tr> 762<td> 763 <p> 764 parameter estimation member functions 765 </p> 766 </td> 767<td> 768 <p> 769 See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 770 Binomial Distribution</a> 771 </p> 772 </td> 773</tr> 774<tr> 775<td> 776 <p> 777 <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code> 778 </p> 779 </td> 780<td> 781 <p> 782 See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 783 Binomial Distribution</a> 784 </p> 785 </td> 786</tr> 787<tr> 788<td> 789 <p> 790 <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code> 791 </p> 792 </td> 793<td> 794 <p> 795 See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 796 Binomial Distribution</a> 797 </p> 798 </td> 799</tr> 800<tr> 801<td> 802 <p> 803 <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code> 804 </p> 805 </td> 806<td> 807 <p> 808 See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 809 Binomial Distribution</a> 810 </p> 811 </td> 812</tr> 813<tr> 814<td> 815 <p> 816 <code class="computeroutput"><span class="identifier">find_maximum_number_of_trials</span></code> 817 </p> 818 </td> 819<td> 820 <p> 821 See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative 822 Binomial Distribution</a> 823 </p> 824 </td> 825</tr> 826</tbody> 827</table></div> 828</div> 829<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 830<td align="left"></td> 831<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 832 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 833 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 834 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 835 Daryle Walker and Xiaogang Zhang<p> 836 Distributed under the Boost Software License, Version 1.0. (See accompanying 837 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>) 838 </p> 839</div></td> 840</tr></table> 841<hr> 842<div class="spirit-nav"> 843<a accesskey="p" href="gamma_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="hyperexponential_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 844</div> 845</body> 846</html> 847
