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| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.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 Binomial Sales Quota Example.</a> |
| </h6></div></div></div> |
| <p> |
| This example program <a href="../../../../../../../../example/negative_binomial_example1.cpp" target="_top">negative_binomial_example1.cpp |
| (full source code)</a> demonstrates a simple use to find the probability |
| of meeting a sales quota. |
| </p> |
| <p> |
| </p> |
| <p> |
| Based on <a href="http://en.wikipedia.org/wiki/Negative_binomial_distribution" target="_top">a |
| problem by Dr. Diane Evans, Professor of Mathematics at Rose-Hulman |
| Institute of Technology</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| Pat is required to sell candy bars to raise money for the 6th grade |
| field trip. There are thirty houses in the neighborhood, and Pat |
| is not supposed to return home until five candy bars have been sold. |
| So the child goes door to door, selling candy bars. At each house, |
| there is a 0.4 probability (40%) of selling one candy bar and a 0.6 |
| probability (60%) of selling nothing. |
| </p> |
| <p> |
| </p> |
| <p> |
| What is the probability mass (density) function (pdf) for selling |
| the last (fifth) candy bar at the nth house? |
| </p> |
| <p> |
| </p> |
| <p> |
| The Negative Binomial(r, p) distribution describes the probability |
| of k failures and r successes in k+r Bernoulli(p) trials with success |
| on the last trial. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli |
| trial</a> is one with only two possible outcomes, success of |
| failure, and p is the probability of success). See also <a href="../../../../../" target="_top">http://en.wikipedia.org/wiki/Bernoulli_distribution |
| Bernoulli distribution</a> and <a href="http://www.math.uah.edu/stat/bernoulli/Introduction.xhtml" target="_top">Bernoulli |
| applications</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| In this example, we will deliberately produce a variety of calculations |
| and outputs to demonstrate the ways that the negative binomial distribution |
| can be implemented with this library: it is also deliberately over-commented. |
| </p> |
| <p> |
| </p> |
| <p> |
| First we need to #define macros to control the error and discrete |
| handling policies. For this simple example, we want to avoid throwing |
| an exception (the default policy) and just return infinity. We want |
| to treat the distribution as if it was continuous, so we choose a |
| discrete_quantile policy of real, rather than the default policy |
| integer_round_outwards. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">ignore_error</span> |
| <span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| After that we need some includes to provide easy access to the negative |
| binomial distribution, |
| </p> |
| <p> |
| </p> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| It is vital to #include distributions etc <span class="bold"><strong>after</strong></span> |
| the above #defines |
| </p></td></tr> |
| </table></div> |
| <p> |
| </p> |
| <p> |
| and we need some std library iostream, of course. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <span class="comment">// for negative_binomial_distribution |
| </span> <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> <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> <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> <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> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <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> |
| <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> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> |
| <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> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| It is always sensible to use try and catch blocks because defaults |
| policies are to throw an exception if anything goes wrong. |
| </p> |
| <p> |
| </p> |
| <p> |
| A simple catch block (see below) will ensure that you get a helpful |
| error message instead of an abrupt program abort. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">try</span> |
| <span class="special">{</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Selling five candy bars means getting five successes, so successes |
| r = 5. The total number of trials (n, in this case, houses visited) |
| this takes is therefore = sucesses + failures or k + r = k + 5. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| At each house, there is a 0.4 probability (40%) of selling one candy |
| bar and a 0.6 probability (60%) of selling nothing. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| The Negative Binomial(r, p) distribution describes the probability |
| of k failures and r successes in k+r Bernoulli(p) trials with success |
| on the last trial. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli |
| trial</a> is one with only two possible outcomes, success of |
| failure, and p is the probability of success). |
| </p> |
| <p> |
| </p> |
| <p> |
| We therefore start by constructing a negative binomial distribution |
| with parameters sales_quota (required successes) and probability |
| of success. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| To confirm, display the success_fraction & successes parameters |
| of the distribution. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="string">", he might expect, on average,\n"</span> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Pat has a sales per house success rate of 0.4. |
| Therefore he would, on average, sell 40 bars after trying 100 houses. |
| With a success rate of 0.4, he might expect, on average, |
| to need to visit about 12 houses in order to sell all 5 bars. |
| </pre> |
| <p> |
| </p> |
| <p> |
| The random variable of interest is the number of houses that must |
| be visited to sell five candy bars, so we substitute k = n - 5 into |
| a negative_binomial(5, 0.4) and obtain the <a class="link" href="../../../dist_ref/nmp.html#math.dist.pdf">probability |
| mass (density) function (pdf or pmf)</a> of the distribution of |
| houses visited. Obviously, the best possible case is that Pat makes |
| sales on all the first five houses. |
| </p> |
| <p> |
| </p> |
| <p> |
| We calculate this using the pdf function: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Of course, he could not finish on fewer than 5 houses because he |
| must sell 5 candy bars. So the 5th house is the first that he could |
| possibly finish on. |
| </p> |
| <p> |
| </p> |
| <p> |
| To finish on or before the 8th house, Pat must finish at the 5th, |
| 6th, 7th or 8th house. The probability that he will finish on <span class="bold"><strong>exactly</strong></span> ( == ) on any house is the Probability |
| Density Function (pdf). |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the 7th 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="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> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability that Pat finishes on the 8th 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="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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability that Pat finishes on the 6th house is 0.03072 |
| Probability that Pat finishes on the 7th house is 0.055296 |
| Probability that Pat finishes on the 8th house is 0.077414 |
| </pre> |
| <p> |
| </p> |
| <p> |
| The sum of the probabilities for these houses is the Cumulative Distribution |
| Function (cdf). We can calculate it by adding the individual probabilities. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <span class="string">"\n"</span> <span class="special"><<</span> <span class="string">"pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = "</span> |
| <span class="comment">// Sum each of the mass/density probabilities for houses sales_quota = 5, 6, 7, & 8. |
| </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">5</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 0 failures. |
| </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">6</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 1 failure. |
| </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">7</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 2 failures. |
| </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">8</span> <span class="special">-</span> <span class="identifier">sales_quota</span><span class="special">)</span> <span class="comment">// 3 failures. |
| </span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">pdf(sales_quota) + pdf(6) + pdf(7) + pdf(8) = 0.17367 |
| </pre> |
| <p> |
| </p> |
| <p> |
| Or, usually better, by using the negative binomial <span class="bold"><strong>cumulative</strong></span> |
| distribution function. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability of selling his quota of 5 bars on or before the 8th house is 0.17367 |
| </pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability that Pat finishes exactly on the 10th house is 0.10033 |
| Probability of selling his quota of 5 bars on or before the 10th house is 0.3669 |
| </pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability that Pat finishes on the 11th house is 0.10033 |
| Probability of selling his quota of 5 candy bars |
| on or before the 11th house is 0.46723 |
| </pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability that Pat finishes on the 12th house is 0.094596 |
| Probability of selling his quota of 5 candy bars |
| on or before the 12th house is 0.56182 |
| </pre> |
| <p> |
| </p> |
| <p> |
| Finally consider the risk of Pat not selling his quota of 5 bars |
| even after visiting all the houses. Calculate the probability that |
| he <span class="emphasis"><em>will</em></span> sell on or before the last house: Calculate |
| the probability that he would sell all his quota on the very last |
| house. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Probability of selling his quota of 5 bars on the 30th house is |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability that Pat finishes on the 30 house is 0.00069145 |
| </pre> |
| <p> |
| </p> |
| <p> |
| when he'd be very unlucky indeed! |
| </p> |
| <p> |
| </p> |
| <p> |
| What is the probability that Pat exhausts all 30 houses in the neighborhood, |
| and <span class="bold"><strong>still</strong></span> doesn't sell the required |
| 5 candy bars? |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability of selling his quota of 5 bars |
| on or before the 30th house is 0.99849 |
| </pre> |
| <p> |
| </p> |
| <p> |
| /*<code class="computeroutput"><span class="identifier">So</span> <span class="identifier">the</span> |
| <span class="identifier">risk</span> <span class="identifier">of</span> |
| <span class="identifier">failing</span> <span class="identifier">even</span> |
| <span class="identifier">after</span> <span class="identifier">visiting</span> |
| <span class="identifier">all</span> <span class="identifier">the</span> |
| <span class="identifier">houses</span> <span class="identifier">is</span> |
| <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> |
| <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></code><code class="computeroutput"> |
| <span class="identifier">But</span> <span class="keyword">using</span> |
| <span class="keyword">this</span> <span class="identifier">expression</span> |
| <span class="identifier">may</span> <span class="identifier">cause</span> |
| <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> |
| <span class="identifier">be</span> <span class="identifier">much</span> |
| <span class="identifier">better</span> <span class="identifier">to</span> |
| <span class="identifier">use</span> <span class="identifier">the</span> |
| <span class="identifier">complement</span> <span class="identifier">of</span> |
| <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> |
| <span class="identifier">of</span> <span class="identifier">failing</span> |
| <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> |
| <span class="identifier">houses</span> <span class="identifier">is</span> |
| <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> |
| <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></code>` But using this expression may cause |
| serious inaccuracy. So it would be much better to use the complement |
| of the cdf. <a class="link" href="../../overview/complements.html#why_complements">Why complements?</a> |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability of failing to sell his quota of 5 bars |
| even after visiting all 30 houses is 0.0015101 |
| </pre> |
| <p> |
| </p> |
| <p> |
| We can also use the quantile (percentile), the inverse of the cdf, |
| to predict which house Pat will finish on. So for the 8th house: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Probability of meeting sales quota on or before 8th house is 0.174 |
| </pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">If the confidence of meeting sales quota is 0.17367, then the finishing house is 8 |
| </pre> |
| <p> |
| </p> |
| <p> |
| Demanding absolute certainty that all 5 will be sold, implies an |
| infinite number of trials. (Of course, there are only 30 houses on |
| the estate, so he can't ever be <span class="bold"><strong>certain</strong></span> |
| of selling his quota). |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <span class="comment">// 1.#INF == infinity.</span></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">If the confidence of meeting sales quota is 1, then the finishing house is 1.#INF |
| </pre> |
| <p> |
| </p> |
| <p> |
| And similarly for a few other probabilities: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| |
| <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> |
| <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> |
| |
| <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> <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">If the confidence of meeting sales quota is 0, then the finishing house is 5 |
| If the confidence of meeting sales quota is 0.5, then the finishing house is 11.337 |
| If the confidence of meeting sales quota is 0.99849, then the finishing house is 30 |
| </pre> |
| <p> |
| </p> |
| <p> |
| Notice that because we chose a discrete quantile policy of real, |
| the result can be an 'unreal' fractional house. |
| </p> |
| <p> |
| </p> |
| <p> |
| If the opposite is true, we don't want to assume any confidence, |
| then this is tantamount to assuming that all the first sales_quota |
| trials will be successful sales. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">If confidence of meeting quota is zero (we assume all houses are successful sales), then finishing house is 5 |
| If confidence of meeting quota is 0, then finishing house is 5 |
| </pre> |
| <p> |
| </p> |
| <p> |
| We can list quantiles for a few probabilities: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <span class="comment">// Confidence as fraction = 1-alpha, as percent = 100 * (1-alpha[i]) % |
| </span> <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> |
| <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> |
| <span class="special">{</span> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">If confidence of meeting quota is 0, then finishing house is 5 |
| If confidence of meeting quota is 0.001, then finishing house is 5 |
| If confidence of meeting quota is 0.01, then finishing house is 5 |
| If confidence of meeting quota is 0.05, then finishing house is 6.2 |
| If confidence of meeting quota is 0.1, then finishing house is 7.06 |
| If confidence of meeting quota is 0.5, then finishing house is 11.3 |
| If confidence of meeting quota is 0.9, then finishing house is 17.8 |
| If confidence of meeting quota is 0.95, then finishing house is 20.1 |
| If confidence of meeting quota is 0.99, then finishing house is 24.8 |
| If confidence of meeting quota is 0.999, then finishing house is 31.1 |
| If confidence of meeting quota is 1, then finishing house is 1.#INF |
| </pre> |
| <p> |
| </p> |
| <p> |
| We could have applied a ceil function to obtain a 'worst case' integer |
| value for house. |
| </p> |
| <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> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Or, if we had used the default discrete quantile policy, integer_outside, |
| by omitting |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span></pre> |
| <p> |
| we would have achieved the same effect. |
| </p> |
| <p> |
| </p> |
| <p> |
| The real result gives some suggestion which house is most likely. |
| For example, compare the real and integer_outside for 95% confidence. |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">If confidence of meeting quota is 0.95, then finishing house is 20.1 |
| If confidence of meeting quota is 0.95, then finishing house is 21 |
| </pre> |
| <p> |
| </p> |
| <p> |
| The real value 20.1 is much closer to 20 than 21, so integer_outside |
| is pessimistic. We could also use integer_round_nearest policy to |
| suggest that 20 is more likely. |
| </p> |
| <p> |
| </p> |
| <p> |
| Finally, we can tabulate the probability for the last sale being |
| exactly on each house. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <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> |
| <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> |
| <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> |
| <span class="special">{</span> |
| <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> |
| <span class="special">}</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">House for 5 th (last) sale. Probability (%) |
| 5 0.01024 |
| 6 0.04096 |
| 7 0.096256 |
| 8 0.17367 |
| 9 0.26657 |
| 10 0.3669 |
| 11 0.46723 |
| 12 0.56182 |
| 13 0.64696 |
| 14 0.72074 |
| 15 0.78272 |
| 16 0.83343 |
| 17 0.874 |
| 18 0.90583 |
| 19 0.93039 |
| 20 0.94905 |
| 21 0.96304 |
| 22 0.97342 |
| 23 0.98103 |
| 24 0.98655 |
| 25 0.99053 |
| 26 0.99337 |
| 27 0.99539 |
| 28 0.99681 |
| 29 0.9978 |
| 30 0.99849 |
| </pre> |
| <p> |
| </p> |
| <p> |
| As noted above, using a catch block is always a good idea, even if |
| you do not expect to use it. |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="special">}</span> |
| <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> |
| <span class="special">{</span> <span class="comment">// Since we have set an overflow policy of ignore_error, |
| </span> <span class="comment">// an overflow exception should never be thrown. |
| </span> <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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| For example, without a ignore domain error policy, if we asked for |
| |
| </p> |
| <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> |
| <p> |
| for example, we would get: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">Message from thrown exception was: |
| Error in function boost::math::pdf(const negative_binomial_distribution<double>&, double): |
| Number of failures argument is -1, but must be >= 0 ! |
| </pre> |
| <p> |
| </p> |
| </div> |
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| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
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