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| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h5 class="title"> | |
| 27 | <a name="math_toolkit.stat_tut.weg.neg_binom_eg.negative_binomial_example1"></a><a class="link" href="negative_binomial_example1.html" title="Negative Binomial Sales Quota Example.">Negative | |
| 28 | Binomial Sales Quota Example.</a> | |
| 29 | </h5></div></div></div> | |
| 30 | <p> | |
| 31 | This example program <a href="../../../../../../example/negative_binomial_example1.cpp" target="_top">negative_binomial_example1.cpp | |
| 32 | (full source code)</a> demonstrates a simple use to find the probability | |
| 33 | of meeting a sales quota. | |
| 34 | </p> | |
| 35 | <p> | |
| 36 | Based on <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution" target="_top">a | |
| 37 | problem by Dr. Diane Evans, Professor of Mathematics at Rose-Hulman Institute | |
| 38 | of Technology</a>. | |
| 39 | </p> | |
| 40 | <p> | |
| 41 | Pat is required to sell candy bars to raise money for the 6th grade field | |
| 42 | trip. There are thirty houses in the neighborhood, and Pat is not supposed | |
| 43 | to return home until five candy bars have been sold. So the child goes | |
| 44 | door to door, selling candy bars. At each house, there is a 0.4 probability | |
| 45 | (40%) of selling one candy bar and a 0.6 probability (60%) of selling | |
| 46 | nothing. | |
| 47 | </p> | |
| 48 | <p> | |
| 49 | What is the probability mass (density) function (pdf) for selling the | |
| 50 | last (fifth) candy bar at the nth house? | |
| 51 | </p> | |
| 52 | <p> | |
| 53 | The Negative Binomial(r, p) distribution describes the probability of | |
| 54 | k failures and r successes in k+r Bernoulli(p) trials with success on | |
| 55 | the last trial. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli | |
| 56 | trial</a> is one with only two possible outcomes, success of failure, | |
| 57 | and p is the probability of success). See also <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli | |
| 58 | distribution</a> and <a href="http://www.math.uah.edu/stat/bernoulli/Introduction.xhtml" target="_top">Bernoulli | |
| 59 | applications</a>. | |
| 60 | </p> | |
| 61 | <p> | |
| 62 | In this example, we will deliberately produce a variety of calculations | |
| 63 | and outputs to demonstrate the ways that the negative binomial distribution | |
| 64 | can be implemented with this library: it is also deliberately over-commented. | |
| 65 | </p> | |
| 66 | <p> | |
| 67 | First we need to #define macros to control the error and discrete handling | |
| 68 | policies. For this simple example, we want to avoid throwing an exception | |
| 69 | (the default policy) and just return infinity. We want to treat the distribution | |
| 70 | as if it was continuous, so we choose a discrete_quantile policy of real, | |
| 71 | rather than the default policy integer_round_outwards. | |
| 72 | </p> | |
| 73 | <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">ignore_error</span> | |
| 74 | <span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span> | |
| 75 | </pre> | |
| 76 | <p> | |
| 77 | After that we need some includes to provide easy access to the negative | |
| 78 | binomial distribution, | |
| 79 | </p> | |
| 80 | <div class="caution"><table border="0" summary="Caution"> | |
| 81 | <tr> | |
| 82 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td> | |
| 83 | <th align="left">Caution</th> | |
| 84 | </tr> | |
| 85 | <tr><td align="left" valign="top"><p> | |
| 86 | It is vital to #include distributions etc <span class="bold"><strong>after</strong></span> | |
| 87 | the above #defines | |
| 88 | </p></td></tr> | |
| 89 | </table></div> | |
| 90 | <p> | |
| 91 | and we need some std library iostream, of course. | |
| 92 | </p> | |
| 93 | <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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 94 | <span class="comment">// for negative_binomial_distribution</span> | |
| 95 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span> | |
| 96 | <span class="keyword">using</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">pdf</span><span class="special">;</span> <span class="comment">// Probability mass function.</span> | |
| 97 | <span class="keyword">using</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">cdf</span><span class="special">;</span> <span class="comment">// Cumulative density function.</span> | |
| 98 | <span class="keyword">using</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">quantile</span><span class="special">;</span> | |
| 99 | ||
| 100 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> | |
| 101 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 102 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> | |
| 103 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> | |
| 104 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> | |
| 105 | ||
| 106 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> | |
| 107 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span> | |
| 108 | </pre> | |
| 109 | <p> | |
| 110 | It is always sensible to use try and catch blocks because defaults policies | |
| 111 | are to throw an exception if anything goes wrong. | |
| 112 | </p> | |
| 113 | <p> | |
| 114 | A simple catch block (see below) will ensure that you get a helpful error | |
| 115 | message instead of an abrupt program abort. | |
| 116 | </p> | |
| 117 | <pre class="programlisting"><span class="keyword">try</span> | |
| 118 | <span class="special">{</span> | |
| 119 | </pre> | |
| 120 | <p> | |
| 121 | Selling five candy bars means getting five successes, so successes r | |
| 122 | = 5. The total number of trials (n, in this case, houses visited) this | |
| 123 | takes is therefore = sucesses + failures or k + r = k + 5. | |
| 124 | </p> | |
| 125 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">sales_quota</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span> <span class="comment">// Pat's sales quota - successes (r).</span> | |
| 126 | </pre> | |
| 127 | <p> | |
| 128 | At each house, there is a 0.4 probability (40%) of selling one candy | |
| 129 | bar and a 0.6 probability (60%) of selling nothing. | |
| 130 | </p> | |
| 131 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">0.4</span><span class="special">;</span> <span class="comment">// success_fraction (p) - so failure_fraction is 0.6.</span> | |
| 132 | </pre> | |
| 133 | <p> | |
| 134 | The Negative Binomial(r, p) distribution describes the probability of | |
| 135 | k failures and r successes in k+r Bernoulli(p) trials with success on | |
| 136 | the last trial. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli | |
| 137 | trial</a> is one with only two possible outcomes, success of failure, | |
| 138 | and p is the probability of success). | |
| 139 | </p> | |
| 140 | <p> | |
| 141 | We therefore start by constructing a negative binomial distribution with | |
| 142 | parameters sales_quota (required successes) and probability of success. | |
| 143 | </p> | |
| 144 | <pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">nb</span><span class="special">(</span><span class="identifier">sales_quota</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> <span class="comment">// type double by default.</span> | |
| 145 | </pre> | |
| 146 | <p> | |
| 147 | To confirm, display the success_fraction & successes parameters of | |
| 148 | the distribution. | |
| 149 | </p> | |
| 150 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Pat has a sales per house success rate of "</span> <span class="special"><<</span> <span class="identifier">success_fraction</span> | |
| 151 | <span class="special"><<</span> <span class="string">".\nTherefore he would, on average, sell "</span> <span class="special"><<</span> <span class="identifier">nb</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special">*</span> <span class="number">100</span> | |
| 152 | <span class="special"><<</span> <span class="string">" bars after trying 100 houses."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 153 | ||
| 154 | <span class="keyword">int</span> <span class="identifier">all_houses</span> <span class="special">=</span> <span class="number">30</span><span class="special">;</span> <span class="comment">// The number of houses on the estate.</span> | |
| 155 | ||
| 156 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"With a success rate of "</span> <span class="special"><<</span> <span class="identifier">nb</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> | |
| 157 | <span class="special"><<</span> <span class="string">", he might expect, on average,\n"</span> | |
| 158 | <span class="string">"to need to visit about "</span> <span class="special"><<</span> <span class="identifier">success_fraction</span> <span class="special">*</span> <span class="identifier">all_houses</span> | |
| 159 | <span class="special"><<</span> <span class="string">" houses in order to sell all "</span> <span class="special"><<</span> <span class="identifier">nb</span><span class="special">.</span><span class="identifier">successes</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" bars. "</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 160 | </pre> | |
| 161 | <pre class="programlisting">Pat has a sales per house success rate of 0.4. | |
| 162 | Therefore he would, on average, sell 40 bars after trying 100 houses. | |
| 163 | With a success rate of 0.4, he might expect, on average, | |
| 164 | to need to visit about 12 houses in order to sell all 5 bars. | |
| 165 | </pre> | |
| 166 | <p> | |
| 167 | The random variable of interest is the number of houses that must be | |
| 168 | visited to sell five candy bars, so we substitute k = n - 5 into a negative_binomial(5, | |
| 169 | 0.4) and obtain the <a class="link" href="../../../dist_ref/nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability | |
| 170 | Density Function</a> of the distribution of houses visited. Obviously, | |
| 171 | the best possible case is that Pat makes sales on all the first five | |
| 172 | houses. | |
| 173 | </p> | |
| 174 | <p> | |
| 175 | We calculate this using the pdf function: | |
| 176 | </p> | |
| 177 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="string">"th house is "</span> | |
| 178 | <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">5</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// == pdf(nb, 0)</span> | |
| 179 | </pre> | |
| 180 | <p> | |
| 181 | Of course, he could not finish on fewer than 5 houses because he must | |
| 182 | sell 5 candy bars. So the 5th house is the first that he could possibly | |
| 183 | finish on. | |
| 184 | </p> | |
| 185 | <p> | |
| 186 | To finish on or before the 8th house, Pat must finish at the 5th, 6th, | |
| 187 | 7th or 8th house. The probability that he will finish on <span class="bold"><strong>exactly</strong></span> | |
| 188 | ( == ) on any house is the Probability Density Function (pdf). | |
| 189 | </p> | |
| 190 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the 6th house is "</span> | |
| 191 | <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">6</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 192 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the 7th house is "</span> | |
| 193 | <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">7</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 194 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the 8th house is "</span> | |
| 195 | <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">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 196 | </pre> | |
| 197 | <pre class="programlisting">Probability that Pat finishes on the 6th house is 0.03072 | |
| 198 | Probability that Pat finishes on the 7th house is 0.055296 | |
| 199 | Probability that Pat finishes on the 8th house is 0.077414 | |
| 200 | </pre> | |
| 201 | <p> | |
| 202 | The sum of the probabilities for these houses is the Cumulative Distribution | |
| 203 | Function (cdf). We can calculate it by adding the individual probabilities. | |
| 204 | </p> | |
| 205 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on or before the 8th house is sum "</span> | |
| 206 | <span class="string">"\n"</span> <span class="special"><<</span> <span class="string">"pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = "</span> | |
| 207 | <span class="comment">// Sum each of the mass/density probabilities for houses sales_quota = 5, 6, 7, & 8.</span> | |
| 208 | <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">5</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 0 failures.</span> | |
| 209 | <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">6</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 1 failure.</span> | |
| 210 | <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">7</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 2 failures.</span> | |
| 211 | <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">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 3 failures.</span> | |
| 212 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 213 | </pre> | |
| 214 | <pre class="programlisting">pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = 0.17367 | |
| 215 | </pre> | |
| 216 | <p> | |
| 217 | Or, usually better, by using the negative binomial <span class="bold"><strong>cumulative</strong></span> | |
| 218 | distribution function. | |
| 219 | </p> | |
| 220 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> | |
| 221 | <span class="special"><<</span> <span class="string">" bars\non or before the "</span> <span class="special"><<</span> <span class="number">8</span> <span class="special"><<</span> <span class="string">"th house is "</span> | |
| 222 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 223 | </pre> | |
| 224 | <pre class="programlisting">Probability of selling his quota of 5 bars on or before the 8th house is 0.17367 | |
| 225 | </pre> | |
| 226 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability that Pat finishes exactly on the 10th house is "</span> | |
| 227 | <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">10</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 228 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> | |
| 229 | <span class="special"><<</span> <span class="string">" bars\non or before the "</span> <span class="special"><<</span> <span class="number">10</span> <span class="special"><<</span> <span class="string">"th house is "</span> | |
| 230 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">10</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 231 | </pre> | |
| 232 | <pre class="programlisting">Probability that Pat finishes exactly on the 10th house is 0.10033 | |
| 233 | Probability of selling his quota of 5 bars on or before the 10th house is 0.3669 | |
| 234 | </pre> | |
| 235 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes exactly on the 11th house is "</span> | |
| 236 | <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">11</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 237 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> | |
| 238 | <span class="special"><<</span> <span class="string">" bars\non or before the "</span> <span class="special"><<</span> <span class="number">11</span> <span class="special"><<</span> <span class="string">"th house is "</span> | |
| 239 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">11</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 240 | </pre> | |
| 241 | <pre class="programlisting">Probability that Pat finishes on the 11th house is 0.10033 | |
| 242 | Probability of selling his quota of 5 candy bars | |
| 243 | on or before the 11th house is 0.46723 | |
| 244 | </pre> | |
| 245 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes exactly on the 12th house is "</span> | |
| 246 | <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">12</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 247 | ||
| 248 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> | |
| 249 | <span class="special"><<</span> <span class="string">" bars\non or before the "</span> <span class="special"><<</span> <span class="number">12</span> <span class="special"><<</span> <span class="string">"th house is "</span> | |
| 250 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">12</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 251 | </pre> | |
| 252 | <pre class="programlisting">Probability that Pat finishes on the 12th house is 0.094596 | |
| 253 | Probability of selling his quota of 5 candy bars | |
| 254 | on or before the 12th house is 0.56182 | |
| 255 | </pre> | |
| 256 | <p> | |
| 257 | Finally consider the risk of Pat not selling his quota of 5 bars even | |
| 258 | after visiting all the houses. Calculate the probability that he <span class="emphasis"><em>will</em></span> | |
| 259 | sell on or before the last house: Calculate the probability that he would | |
| 260 | sell all his quota on the very last house. | |
| 261 | </p> | |
| 262 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the "</span> <span class="special"><<</span> <span class="identifier">all_houses</span> | |
| 263 | <span class="special"><<</span> <span class="string">" house is "</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="identifier">all_houses</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 264 | </pre> | |
| 265 | <p> | |
| 266 | Probability of selling his quota of 5 bars on the 30th house is | |
| 267 | </p> | |
| 268 | <pre class="programlisting">Probability that Pat finishes on the 30 house is 0.00069145 | |
| 269 | </pre> | |
| 270 | <p> | |
| 271 | when he'd be very unlucky indeed! | |
| 272 | </p> | |
| 273 | <p> | |
| 274 | What is the probability that Pat exhausts all 30 houses in the neighborhood, | |
| 275 | and <span class="bold"><strong>still</strong></span> doesn't sell the required | |
| 276 | 5 candy bars? | |
| 277 | </p> | |
| 278 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability of selling his quota of "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> | |
| 279 | <span class="special"><<</span> <span class="string">" bars\non or before the "</span> <span class="special"><<</span> <span class="identifier">all_houses</span> <span class="special"><<</span> <span class="string">"th house is "</span> | |
| 280 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 281 | </pre> | |
| 282 | <pre class="programlisting">Probability of selling his quota of 5 bars | |
| 283 | on or before the 30th house is 0.99849 | |
| 284 | </pre> | |
| 285 | <p> | |
| 286 | /*<code class="computeroutput"><span class="identifier">So</span> <span class="identifier">the</span> | |
| 287 | <span class="identifier">risk</span> <span class="identifier">of</span> | |
| 288 | <span class="identifier">failing</span> <span class="identifier">even</span> | |
| 289 | <span class="identifier">after</span> <span class="identifier">visiting</span> | |
| 290 | <span class="identifier">all</span> <span class="identifier">the</span> | |
| 291 | <span class="identifier">houses</span> <span class="identifier">is</span> | |
| 292 | <span class="number">1</span> <span class="special">-</span> <span class="keyword">this</span> <span class="identifier">probability</span><span class="special">,</span> </code><code class="computeroutput"><span class="number">1</span> | |
| 293 | <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> | |
| 294 | <span class="special">-</span> <span class="identifier">sales_quota</span></code><code class="computeroutput"> | |
| 295 | <span class="identifier">But</span> <span class="keyword">using</span> | |
| 296 | <span class="keyword">this</span> <span class="identifier">expression</span> | |
| 297 | <span class="identifier">may</span> <span class="identifier">cause</span> | |
| 298 | <span class="identifier">serious</span> <span class="identifier">inaccuracy</span><span class="special">,</span> <span class="identifier">so</span> <span class="identifier">it</span> <span class="identifier">would</span> | |
| 299 | <span class="identifier">be</span> <span class="identifier">much</span> | |
| 300 | <span class="identifier">better</span> <span class="identifier">to</span> | |
| 301 | <span class="identifier">use</span> <span class="identifier">the</span> | |
| 302 | <span class="identifier">complement</span> <span class="identifier">of</span> | |
| 303 | <span class="identifier">the</span> <span class="identifier">cdf</span><span class="special">:</span> <span class="identifier">So</span> <span class="identifier">the</span> <span class="identifier">risk</span> | |
| 304 | <span class="identifier">of</span> <span class="identifier">failing</span> | |
| 305 | <span class="identifier">even</span> <span class="identifier">at</span><span class="special">,</span> <span class="keyword">or</span> <span class="identifier">after</span><span class="special">,</span> <span class="identifier">the</span> <span class="number">31</span><span class="identifier">th</span> <span class="special">(</span><span class="identifier">non</span><span class="special">-</span><span class="identifier">existent</span><span class="special">)</span> | |
| 306 | <span class="identifier">houses</span> <span class="identifier">is</span> | |
| 307 | <span class="number">1</span> <span class="special">-</span> <span class="keyword">this</span> <span class="identifier">probability</span><span class="special">,</span> </code><code class="computeroutput"><span class="number">1</span> | |
| 308 | <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> | |
| 309 | <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span></code>` But using this expression may cause | |
| 310 | serious inaccuracy. So it would be much better to use the __complement | |
| 311 | of the cdf (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>). | |
| 312 | </p> | |
| 313 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nProbability of failing to sell his quota of "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> | |
| 314 | <span class="special"><<</span> <span class="string">" bars\neven after visiting all "</span> <span class="special"><<</span> <span class="identifier">all_houses</span> <span class="special"><<</span> <span class="string">" houses is "</span> | |
| 315 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">all_houses</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 316 | </pre> | |
| 317 | <pre class="programlisting">Probability of failing to sell his quota of 5 bars | |
| 318 | even after visiting all 30 houses is 0.0015101 | |
| 319 | </pre> | |
| 320 | <p> | |
| 321 | We can also use the quantile (percentile), the inverse of the cdf, to | |
| 322 | predict which house Pat will finish on. So for the 8th house: | |
| 323 | </p> | |
| 324 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="special">(</span><span class="number">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">));</span> | |
| 325 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of meeting sales quota on or before 8th house is "</span><span class="special"><<</span> <span class="identifier">p</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 326 | </pre> | |
| 327 | <pre class="programlisting">Probability of meeting sales quota on or before 8th house is 0.174 | |
| 328 | </pre> | |
| 329 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special"><<</span> <span class="identifier">p</span> | |
| 330 | <span class="special"><<</span> <span class="string">", then the finishing house is "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 331 | ||
| 332 | <span class="identifier">cout</span><span class="special"><<</span> <span class="string">" quantile(nb, p) = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 333 | </pre> | |
| 334 | <pre class="programlisting">If the confidence of meeting sales quota is 0.17367, then the finishing house is 8 | |
| 335 | </pre> | |
| 336 | <p> | |
| 337 | Demanding absolute certainty that all 5 will be sold, implies an infinite | |
| 338 | number of trials. (Of course, there are only 30 houses on the estate, | |
| 339 | so he can't ever be <span class="bold"><strong>certain</strong></span> of selling | |
| 340 | his quota). | |
| 341 | </p> | |
| 342 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special"><<</span> <span class="number">1.</span> | |
| 343 | <span class="special"><<</span> <span class="string">", then the finishing house is "</span> <span class="special"><<</span> <span class="identifier">quantile</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">sales_quota</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 344 | <span class="comment">// 1.#INF == infinity.</span> | |
| 345 | </pre> | |
| 346 | <pre class="programlisting">If the confidence of meeting sales quota is 1, then the finishing house is 1.#INF | |
| 347 | </pre> | |
| 348 | <p> | |
| 349 | And similarly for a few other probabilities: | |
| 350 | </p> | |
| 351 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special"><<</span> <span class="number">0.</span> | |
| 352 | <span class="special"><<</span> <span class="string">", then the finishing house is "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">0.</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 353 | ||
| 354 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special"><<</span> <span class="number">0.5</span> | |
| 355 | <span class="special"><<</span> <span class="string">", then the finishing house is "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 356 | ||
| 357 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If the confidence of meeting sales quota is "</span> <span class="special"><<</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.00151</span> <span class="comment">// 30 th</span> | |
| 358 | <span class="special"><<</span> <span class="string">", then the finishing house is "</span> <span class="special"><<</span> <span class="identifier">quantile</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="number">0.00151</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 359 | </pre> | |
| 360 | <pre class="programlisting">If the confidence of meeting sales quota is 0, then the finishing house is 5 | |
| 361 | If the confidence of meeting sales quota is 0.5, then the finishing house is 11.337 | |
| 362 | If the confidence of meeting sales quota is 0.99849, then the finishing house is 30 | |
| 363 | </pre> | |
| 364 | <p> | |
| 365 | Notice that because we chose a discrete quantile policy of real, the | |
| 366 | result can be an 'unreal' fractional house. | |
| 367 | </p> | |
| 368 | <p> | |
| 369 | If the opposite is true, we don't want to assume any confidence, then | |
| 370 | this is tantamount to assuming that all the first sales_quota trials | |
| 371 | will be successful sales. | |
| 372 | </p> | |
| 373 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If confidence of meeting quota is zero\n(we assume all houses are successful sales)"</span> | |
| 374 | <span class="string">", then finishing house is "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 375 | </pre> | |
| 376 | <pre class="programlisting">If confidence of meeting quota is zero (we assume all houses are successful sales), then finishing house is 5 | |
| 377 | If confidence of meeting quota is 0, then finishing house is 5 | |
| 378 | </pre> | |
| 379 | <p> | |
| 380 | We can list quantiles for a few probabilities: | |
| 381 | </p> | |
| 382 | <pre class="programlisting"> <span class="keyword">double</span> <span class="identifier">ps</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span><span class="number">0.</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.9</span><span class="special">,</span> <span class="number">0.95</span><span class="special">,</span> <span class="number">0.99</span><span class="special">,</span> <span class="number">0.999</span><span class="special">,</span> <span class="number">1.</span><span class="special">};</span> | |
| 383 | <span class="comment">// Confidence as fraction = 1-alpha, as percent = 100 * (1-alpha[i]) %</span> | |
| 384 | <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">3</span><span class="special">);</span> | |
| 385 | <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">ps</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">ps</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="identifier">i</span><span class="special">++)</span> | |
| 386 | <span class="special">{</span> | |
| 387 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If confidence of meeting quota is "</span> <span class="special"><<</span> <span class="identifier">ps</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> | |
| 388 | <span class="special"><<</span> <span class="string">", then finishing house is "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">ps</span><span class="special">[</span><span class="identifier">i</span><span class="special">])</span> <span class="special">+</span> <span class="identifier">sales_quota</span> | |
| 389 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 390 | <span class="special">}</span> | |
| 391 | </pre> | |
| 392 | <pre class="programlisting">If confidence of meeting quota is 0, then finishing house is 5 | |
| 393 | If confidence of meeting quota is 0.001, then finishing house is 5 | |
| 394 | If confidence of meeting quota is 0.01, then finishing house is 5 | |
| 395 | If confidence of meeting quota is 0.05, then finishing house is 6.2 | |
| 396 | If confidence of meeting quota is 0.1, then finishing house is 7.06 | |
| 397 | If confidence of meeting quota is 0.5, then finishing house is 11.3 | |
| 398 | If confidence of meeting quota is 0.9, then finishing house is 17.8 | |
| 399 | If confidence of meeting quota is 0.95, then finishing house is 20.1 | |
| 400 | If confidence of meeting quota is 0.99, then finishing house is 24.8 | |
| 401 | If confidence of meeting quota is 0.999, then finishing house is 31.1 | |
| 402 | If confidence of meeting quota is 1, then finishing house is 1.#INF | |
| 403 | </pre> | |
| 404 | <p> | |
| 405 | We could have applied a ceil function to obtain a 'worst case' integer | |
| 406 | value for house. | |
| 407 | </p> | |
| 408 | <pre class="programlisting"><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">ps</span><span class="special">[</span><span class="identifier">i</span><span class="special">]))</span></pre> | |
| 409 | <p> | |
| 410 | Or, if we had used the default discrete quantile policy, integer_outside, | |
| 411 | by omitting | |
| 412 | </p> | |
| 413 | <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span></pre> | |
| 414 | <p> | |
| 415 | we would have achieved the same effect. | |
| 416 | </p> | |
| 417 | <p> | |
| 418 | The real result gives some suggestion which house is most likely. For | |
| 419 | example, compare the real and integer_outside for 95% confidence. | |
| 420 | </p> | |
| 421 | <pre class="programlisting">If confidence of meeting quota is 0.95, then finishing house is 20.1 | |
| 422 | If confidence of meeting quota is 0.95, then finishing house is 21 | |
| 423 | </pre> | |
| 424 | <p> | |
| 425 | The real value 20.1 is much closer to 20 than 21, so integer_outside | |
| 426 | is pessimistic. We could also use integer_round_nearest policy to suggest | |
| 427 | that 20 is more likely. | |
| 428 | </p> | |
| 429 | <p> | |
| 430 | Finally, we can tabulate the probability for the last sale being exactly | |
| 431 | on each house. | |
| 432 | </p> | |
| 433 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nHouse for "</span> <span class="special"><<</span> <span class="identifier">sales_quota</span> <span class="special"><<</span> <span class="string">"th (last) sale. Probability (%)"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 434 | <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span> | |
| 435 | <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">sales_quota</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="identifier">all_houses</span><span class="special">+</span><span class="number">1</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++)</span> | |
| 436 | <span class="special">{</span> | |
| 437 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">8</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="identifier">i</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 438 | <span class="special">}</span> | |
| 439 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 440 | </pre> | |
| 441 | <pre class="programlisting">House for 5 th (last) sale. Probability (%) | |
| 442 | 5 0.01024 | |
| 443 | 6 0.04096 | |
| 444 | 7 0.096256 | |
| 445 | 8 0.17367 | |
| 446 | 9 0.26657 | |
| 447 | 10 0.3669 | |
| 448 | 11 0.46723 | |
| 449 | 12 0.56182 | |
| 450 | 13 0.64696 | |
| 451 | 14 0.72074 | |
| 452 | 15 0.78272 | |
| 453 | 16 0.83343 | |
| 454 | 17 0.874 | |
| 455 | 18 0.90583 | |
| 456 | 19 0.93039 | |
| 457 | 20 0.94905 | |
| 458 | 21 0.96304 | |
| 459 | 22 0.97342 | |
| 460 | 23 0.98103 | |
| 461 | 24 0.98655 | |
| 462 | 25 0.99053 | |
| 463 | 26 0.99337 | |
| 464 | 27 0.99539 | |
| 465 | 28 0.99681 | |
| 466 | 29 0.9978 | |
| 467 | 30 0.99849 | |
| 468 | </pre> | |
| 469 | <p> | |
| 470 | As noted above, using a catch block is always a good idea, even if you | |
| 471 | do not expect to use it. | |
| 472 | </p> | |
| 473 | <pre class="programlisting"><span class="special">}</span> | |
| 474 | <span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&</span> <span class="identifier">e</span><span class="special">)</span> | |
| 475 | <span class="special">{</span> <span class="comment">// Since we have set an overflow policy of ignore_error,</span> | |
| 476 | <span class="comment">// an overflow exception should never be thrown.</span> | |
| 477 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nMessage from thrown exception was:\n "</span> <span class="special"><<</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 478 | </pre> | |
| 479 | <p> | |
| 480 | For example, without a ignore domain error policy, if we asked for | |
| 481 | </p> | |
| 482 | <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span></pre> | |
| 483 | <p> | |
| 484 | for example, we would get: | |
| 485 | </p> | |
| 486 | <pre class="programlisting">Message from thrown exception was: | |
| 487 | Error in function boost::math::pdf(const negative_binomial_distribution<double>&, double): | |
| 488 | Number of failures argument is -1, but must be >= 0 ! | |
| 489 | </pre> | |
| 490 | </div> | |
| 491 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 492 | <td align="left"></td> | |
| 493 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 494 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 495 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 496 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 497 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 498 | 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>) | |
| 499 | </p> | |
| 500 | </div></td> | |
| 501 | </tr></table> | |
| 502 | <hr> | |
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