1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Binomial Quiz Example</title> 5<link rel="stylesheet" href="../../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../binom_eg.html" title="Binomial Distribution Examples"> 9<link rel="prev" href="binomial_coinflip_example.html" title="Binomial Coin-Flipping Example"> 10<link rel="next" href="binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="binomial_coinflip_example.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../binom_eg.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="binom_conf.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h5 class="title"> 27<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example"></a><a class="link" href="binomial_quiz_example.html" title="Binomial Quiz Example">Binomial 28 Quiz Example</a> 29</h5></div></div></div> 30<p> 31 A multiple choice test has four possible answers to each of 16 questions. 32 A student guesses the answer to each question, so the probability of 33 getting a correct answer on any given question is one in four, a quarter, 34 1/4, 25% or fraction 0.25. The conditions of the binomial experiment 35 are assumed to be met: n = 16 questions constitute the trials; each question 36 results in one of two possible outcomes (correct or incorrect); the probability 37 of being correct is 0.25 and is constant if no knowledge about the subject 38 is assumed; the questions are answered independently if the student's 39 answer to a question in no way influences his/her answer to another question. 40 </p> 41<p> 42 First, we need to be able to use the binomial distribution constructor 43 (and some std input/output, of course). 44 </p> 45<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 46 <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">binomial</span><span class="special">;</span> 47 48<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> 49 <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> 50 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ios</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">flush</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="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">fixed</span><span class="special">;</span> 51<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> 52 <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="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> 53<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">exception</span><span class="special">></span> 54</pre> 55<p> 56 The number of correct answers, X, is distributed as a binomial random 57 variable with binomial distribution parameters: questions n and success 58 fraction probability p. So we construct a binomial distribution: 59 </p> 60<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">questions</span> <span class="special">=</span> <span class="number">16</span><span class="special">;</span> <span class="comment">// All the questions in the quiz.</span> 61<span class="keyword">int</span> <span class="identifier">answers</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="comment">// Possible answers to each question.</span> 62<span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">answers</span><span class="special">;</span> <span class="comment">// If a random guess, p = 1/4 = 0.25.</span> 63<span class="identifier">binomial</span> <span class="identifier">quiz</span><span class="special">(</span><span class="identifier">questions</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> 64</pre> 65<p> 66 and display the distribution parameters we used thus: 67 </p> 68<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"In a quiz with "</span> <span class="special"><<</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span> 69 <span class="special"><<</span> <span class="string">" questions and with a probability of guessing right of "</span> 70 <span class="special"><<</span> <span class="identifier">quiz</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">" %"</span> 71 <span class="special"><<</span> <span class="string">" or 1 in "</span> <span class="special"><<</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="number">1.</span> <span class="special">/</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">())</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 72</pre> 73<p> 74 Show a few probabilities of just guessing: 75 </p> 76<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting none right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.010023</span> 77<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly one right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 78<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly two right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 79<span class="keyword">int</span> <span class="identifier">pass_score</span> <span class="special">=</span> <span class="number">11</span><span class="special">;</span> 80<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> <span class="special"><<</span> <span class="string">" answers right by chance is "</span> 81 <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 82<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting all "</span> <span class="special"><<</span> <span class="identifier">questions</span> <span class="special"><<</span> <span class="string">" answers right by chance is "</span> 83 <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">questions</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 84</pre> 85<pre class="programlisting">Probability of getting none right is 0.0100226 86Probability of getting exactly one right is 0.0534538 87Probability of getting exactly two right is 0.133635 88Probability of getting exactly 11 right is 0.000247132 89Probability of getting exactly all 16 answers right by chance is 2.32831e-010 90</pre> 91<p> 92 These don't give any encouragement to guessers! 93 </p> 94<p> 95 We can tabulate the 'getting exactly right' ( == ) probabilities thus: 96 </p> 97<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="string">"Guessed Probability"</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 98<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special"><=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span> 99<span class="special">{</span> 100 <span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span> 101 <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">successes</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">probability</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 102<span class="special">}</span> 103<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 104</pre> 105<pre class="programlisting">Guessed Probability 106 0 0.0100226 107 1 0.0534538 108 2 0.133635 109 3 0.207876 110 4 0.225199 111 5 0.180159 112 6 0.110097 113 7 0.0524273 114 8 0.0196602 115 9 0.00582526 11610 0.00135923 11711 0.000247132 11812 3.43239e-005 11913 3.5204e-006 12014 2.51457e-007 12115 1.11759e-008 12216 2.32831e-010 123</pre> 124<p> 125 Then we can add the probabilities of some 'exactly right' like this: 126 </p> 127<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 128</pre> 129<pre class="programlisting">Probability of getting none or one right is 0.0634764 130</pre> 131<p> 132 But if more than a couple of scores are involved, it is more convenient 133 (and may be more accurate) to use the Cumulative Distribution Function 134 (cdf) instead: 135 </p> 136<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 137</pre> 138<pre class="programlisting">Probability of getting none or one right is 0.0634764 139</pre> 140<p> 141 Since the cdf is inclusive, we can get the probability of getting up 142 to 10 right ( <= ) 143 </p> 144<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting <= 10 right (to fail) is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 145</pre> 146<pre class="programlisting">Probability of getting <= 10 right (to fail) is 0.999715 147</pre> 148<p> 149 To get the probability of getting 11 or more right (to pass), it is tempting 150 to use 151 </p> 152<pre class="programlisting"><span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span></pre> 153<p> 154 to get the probability of > 10 155 </p> 156<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting > 10 right (to pass) is "</span> <span class="special"><<</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 157</pre> 158<pre class="programlisting">Probability of getting > 10 right (to pass) is 0.000285239 159</pre> 160<p> 161 But this should be resisted in favor of using the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a> 162 function (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>). 163 </p> 164<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting > 10 right (to pass) 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">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 165</pre> 166<pre class="programlisting">Probability of getting > 10 right (to pass) is 0.000285239 167</pre> 168<p> 169 And we can check that these two, <= 10 and > 10, add up to unity. 170 </p> 171<pre class="programlisting"><span class="identifier">BOOST_ASSERT</span><span class="special">((</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</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">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)))</span> <span class="special">==</span> <span class="number">1.</span><span class="special">);</span> 172</pre> 173<p> 174 If we want a < rather than a <= test, because the CDF is inclusive, 175 we must subtract one from the score. 176 </p> 177<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting less than "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> 178 <span class="special"><<</span> <span class="string">" (< "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> <span class="special"><<</span> <span class="string">") answers right by guessing is "</span> 179 <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 180</pre> 181<pre class="programlisting">Probability of getting less than 11 (< 11) answers right by guessing is 0.999715 182</pre> 183<p> 184 and similarly to get a >= rather than a > test we also need to 185 subtract one from the score (and can again check the sum is unity). This 186 is because if the cdf is <span class="emphasis"><em>inclusive</em></span>, then its complement 187 must be <span class="emphasis"><em>exclusive</em></span> otherwise there would be one possible 188 outcome counted twice! 189 </p> 190<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting at least "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> 191 <span class="special"><<</span> <span class="string">"(>= "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> <span class="special"><<</span> <span class="string">") answers right by guessing is "</span> 192 <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">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">))</span> 193 <span class="special"><<</span> <span class="string">", only 1 in "</span> <span class="special"><<</span> <span class="number">1</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">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 194 195<span class="identifier">BOOST_ASSERT</span><span class="special">((</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span> <span class="special">-</span><span class="number">1</span><span class="special">)</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">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">)))</span> <span class="special">==</span> <span class="number">1</span><span class="special">);</span> 196</pre> 197<pre class="programlisting">Probability of getting at least 11 (>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83 198</pre> 199<p> 200 Finally we can tabulate some probabilities: 201 </p> 202<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="string">"At most (<=)"</span><span class="string">"\n"</span><span class="string">"Guessed OK Probability"</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 203<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">score</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">score</span> <span class="special"><=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">score</span><span class="special">++)</span> 204<span class="special">{</span> 205 <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">score</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> 206 <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">score</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 207<span class="special">}</span> 208<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 209</pre> 210<pre class="programlisting">At most (<=) 211Guessed OK Probability 212 0 0.01002259576 213 1 0.0634764398 214 2 0.1971110499 215 3 0.4049871101 216 4 0.6301861752 217 5 0.8103454274 218 6 0.9204427481 219 7 0.9728700437 220 8 0.9925302796 221 9 0.9983555346 22210 0.9997147608 22311 0.9999618928 22412 0.9999962167 22513 0.9999997371 22614 0.9999999886 22715 0.9999999998 22816 1 229</pre> 230<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="string">"At least (>)"</span><span class="string">"\n"</span><span class="string">"Guessed OK Probability"</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 231<span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">score</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">score</span> <span class="special"><=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">score</span><span class="special">++)</span> 232<span class="special">{</span> 233 <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">score</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> 234 <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">quiz</span><span class="special">,</span> <span class="identifier">score</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 235<span class="special">}</span> 236</pre> 237<pre class="programlisting">At least (>) 238Guessed OK Probability 239 0 0.9899774042 240 1 0.9365235602 241 2 0.8028889501 242 3 0.5950128899 243 4 0.3698138248 244 5 0.1896545726 245 6 0.07955725188 246 7 0.02712995629 247 8 0.00746972044 248 9 0.001644465374 24910 0.0002852391917 25011 3.810715862e-005 25112 3.783265129e-006 25213 2.628657967e-007 25314 1.140870154e-008 25415 2.328306437e-010 25516 0 256</pre> 257<p> 258 We now consider the probabilities of <span class="bold"><strong>ranges</strong></span> 259 of correct guesses. 260 </p> 261<p> 262 First, calculate the probability of getting a range of guesses right, 263 by adding the exact probabilities of each from low ... high. 264 </p> 265<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">low</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span> <span class="comment">// Getting at least 3 right.</span> 266<span class="keyword">int</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span> <span class="comment">// Getting as most 5 right.</span> 267<span class="keyword">double</span> <span class="identifier">sum</span> <span class="special">=</span> <span class="number">0.</span><span class="special">;</span> 268<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="identifier">low</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><=</span> <span class="identifier">high</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++)</span> 269<span class="special">{</span> 270 <span class="identifier">sum</span> <span class="special">+=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">i</span><span class="special">);</span> 271<span class="special">}</span> 272<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">4</span><span class="special">);</span> 273<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> 274 <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> 275 <span class="special"><<</span> <span class="identifier">sum</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.61323</span> 276</pre> 277<pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132 278</pre> 279<p> 280 Or, usually better, we can use the difference of cdfs instead: 281 </p> 282<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> 283 <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.61323</span> 284</pre> 285<pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132 286</pre> 287<p> 288 And we can also try a few more combinations of high and low choices: 289 </p> 290<pre class="programlisting"><span class="identifier">low</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span> 291<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> 292 <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 and 6 P= 0.91042</span> 293<span class="identifier">low</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">8</span><span class="special">;</span> 294<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> 295 <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 <= x 8 P = 0.9825</span> 296<span class="identifier">low</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> 297<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> 298 <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 4 <= x 4 P = 0.22520</span> 299</pre> 300<pre class="programlisting">Probability of getting between 1 and 6 answers right by guessing is 0.9104 301Probability of getting between 1 and 8 answers right by guessing is 0.9825 302Probability of getting between 4 and 4 answers right by guessing is 0.2252 303</pre> 304<h5> 305<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.h0"></a> 306 <span class="phrase"><a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.using_binomial_distribution_mome"></a></span><a class="link" href="binomial_quiz_example.html#math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.using_binomial_distribution_mome">Using 307 Binomial distribution moments</a> 308 </h5> 309<p> 310 Using moments of the distribution, we can say more about the spread of 311 results from guessing. 312 </p> 313<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"By guessing, on average, one can expect to get "</span> <span class="special"><<</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" correct answers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 314<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviation is "</span> <span class="special"><<</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 315<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span> 316 <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" and "</span> 317 <span class="special"><<</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" correct."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 318<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Mode (the most frequent) is "</span> <span class="special"><<</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 319<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Skewness is "</span> <span class="special"><<</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 320</pre> 321<pre class="programlisting">By guessing, on average, one can expect to get 4 correct answers. 322Standard deviation is 1.732 323So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct. 324Mode (the most frequent) is 4 325Skewness is 0.2887 326</pre> 327<h5> 328<a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.h1"></a> 329 <span class="phrase"><a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.quantiles"></a></span><a class="link" href="binomial_quiz_example.html#math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.quantiles">Quantiles</a> 330 </h5> 331<p> 332 The quantiles (percentiles or percentage points) for a few probability 333 levels: 334 </p> 335<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quartiles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 336 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles</span> 337<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"1 standard deviation "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.33</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 338 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.67</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 sd</span> 339<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Deciles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.1</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 340 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.1</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles</span> 341<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"5 to 95% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 342 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5 to 95%</span> 343<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2.5 to 97.5% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 344 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.025</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.5 to 97.5%</span> 345<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2 to 98% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.02</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 346 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.02</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2 to 98%</span> 347 348<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If guessing then percentiles 1 to 99% will get "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> 349 <span class="special"><<</span> <span class="string">" to "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" right."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 350</pre> 351<p> 352 Notice that these output integral values because the default policy is 353 <code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>. 354 </p> 355<pre class="programlisting">Quartiles 2 to 5 3561 standard deviation 2 to 5 357Deciles 1 to 6 3585 to 95% 0 to 7 3592.5 to 97.5% 0 to 8 3602 to 98% 0 to 8 361</pre> 362<p> 363 Quantiles values are controlled by the <a class="link" href="../../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">understanding 364 discrete quantiles</a> quantile policy chosen. The default is <code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>, so the lower 365 quantile is rounded down, and the upper quantile is rounded up. 366 </p> 367<p> 368 But we might believe that the real values tell us a little more - see 369 <a class="link" href="../../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">discrete functions</a>. 370 </p> 371<p> 372 We could control the policy for <span class="bold"><strong>all</strong></span> 373 distributions by 374 </p> 375<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span> 376 377<span class="identifier">at</span> <span class="identifier">the</span> <span class="identifier">head</span> <span class="identifier">of</span> <span class="identifier">the</span> <span class="identifier">program</span> <span class="identifier">would</span> <span class="identifier">make</span> <span class="keyword">this</span> <span class="identifier">policy</span> <span class="identifier">apply</span> 378</pre> 379<p> 380 to this <span class="bold"><strong>one, and only</strong></span>, translation unit. 381 </p> 382<p> 383 Or we can now create a (typedef for) policy that has discrete quantiles 384 real (here avoiding any 'using namespaces ...' statements): 385 </p> 386<pre class="programlisting"><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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span> 387<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">policies</span><span class="special">::</span><span class="identifier">discrete_quantile</span><span class="special">;</span> 388<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">policies</span><span class="special">::</span><span class="identifier">real</span><span class="special">;</span> 389<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">policies</span><span class="special">::</span><span class="identifier">integer_round_outwards</span><span class="special">;</span> <span class="comment">// Default.</span> 390<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">real</span><span class="special">></span> <span class="special">></span> <span class="identifier">real_quantile_policy</span><span class="special">;</span> 391</pre> 392<p> 393 Add a custom binomial distribution called 394 </p> 395<pre class="programlisting"><span class="identifier">real_quantile_binomial</span></pre> 396<p> 397 that uses 398 </p> 399<pre class="programlisting"><span class="identifier">real_quantile_policy</span></pre> 400<pre class="programlisting"><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">binomial_distribution</span><span class="special">;</span> 401<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">real_quantile_policy</span><span class="special">></span> <span class="identifier">real_quantile_binomial</span><span class="special">;</span> 402</pre> 403<p> 404 Construct an object of this custom distribution: 405 </p> 406<pre class="programlisting"><span class="identifier">real_quantile_binomial</span> <span class="identifier">quiz_real</span><span class="special">(</span><span class="identifier">questions</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> 407</pre> 408<p> 409 And use this to show some quantiles - that now have real rather than 410 integer values. 411 </p> 412<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quartiles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 413 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.25</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles 2 to 4.6212</span> 414<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"1 standard deviation "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.33</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 415 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.67</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 sd 2.6654 4.194</span> 416<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Deciles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.1</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 417 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.1</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles 1.3487 5.7583</span> 418<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"5 to 95% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 419 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5 to 95% 0.83739 6.4559</span> 420<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2.5 to 97.5% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 421 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.025</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.5 to 97.5% 0.42806 7.0688</span> 422<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2 to 98% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.02</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> 423 <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.02</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2 to 98% 0.31311 7.7880</span> 424 425<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If guessing, then percentiles 1 to 99% will get "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> 426 <span class="special"><<</span> <span class="string">" to "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" right."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> 427</pre> 428<pre class="programlisting">Real Quantiles 429Quartiles 2 to 4.621 4301 standard deviation 2.665 to 4.194 431Deciles 1.349 to 5.758 4325 to 95% 0.8374 to 6.456 4332.5 to 97.5% 0.4281 to 7.069 4342 to 98% 0.3131 to 7.252 435If guessing then percentiles 1 to 99% will get 0 to 7.788 right. 436</pre> 437<p> 438 See <a href="../../../../../../example/binomial_quiz_example.cpp" target="_top">binomial_quiz_example.cpp</a> 439 for full source code and output. 440 </p> 441</div> 442<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 443<td align="left"></td> 444<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 445 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 446 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 447 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 448 Daryle Walker and Xiaogang Zhang<p> 449 Distributed under the Boost Software License, Version 1.0. (See accompanying 450 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>) 451 </p> 452</div></td> 453</tr></table> 454<hr> 455<div class="spirit-nav"> 456<a accesskey="p" href="binomial_coinflip_example.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../binom_eg.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="binom_conf.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> 457</div> 458</body> 459</html> 460
