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  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">&lt;</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">&gt;</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">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</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">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</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">&lt;</span><span class="identifier">exception</span><span class="special">&gt;</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">&lt;&lt;</span> <span class="string">"In a quiz with "</span> <span class="special">&lt;&lt;</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span>
  69   <span class="special">&lt;&lt;</span> <span class="string">" questions and with a probability of guessing right of "</span>
  70   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" %"</span>
  71   <span class="special">&lt;&lt;</span> <span class="string">" or 1 in "</span> <span class="special">&lt;&lt;</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">int</span><span class="special">&gt;(</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting none right is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting exactly one right is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  78 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting exactly two right is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting exactly "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by chance is "</span>
  81   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  82 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting all "</span> <span class="special">&lt;&lt;</span> <span class="identifier">questions</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by chance is "</span>
  83   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  84 </pre>
  85 <pre class="programlisting">Probability of getting none right is 0.0100226
  86 Probability of getting exactly one right is 0.0534538
  87 Probability of getting exactly two right is 0.133635
  88 Probability of getting exactly 11 right is 0.000247132
  89 Probability 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">&lt;&lt;</span> <span class="string">"\n"</span> <span class="string">"Guessed Probability"</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</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">&lt;=</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">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">successes</span> <span class="special">&lt;&lt;</span> <span class="string">"      "</span> <span class="special">&lt;&lt;</span> <span class="identifier">probability</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 102 <span class="special">}</span>
 103 <span class="identifier">cout</span> <span class="special">&lt;&lt;</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
 116 10      0.00135923
 117 11      0.000247132
 118 12      3.43239e-005
 119 13      3.5204e-006
 120 14      2.51457e-007
 121 15      1.11759e-008
 122 16      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">&lt;&lt;</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</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 ( &lt;= )
 143           </p>
 144 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting &lt;= 10 right (to fail) is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 145 </pre>
 146 <pre class="programlisting">Probability of getting &lt;= 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 &gt; 10
 155           </p>
 156 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting &gt; 10 right (to pass) is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 157 </pre>
 158 <pre class="programlisting">Probability of getting &gt; 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">&lt;&lt;</span> <span class="string">"Probability of getting &gt; 10 right (to pass) is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 165 </pre>
 166 <pre class="programlisting">Probability of getting &gt; 10 right (to pass) is 0.000285239
 167 </pre>
 168 <p>
 169             And we can check that these two, &lt;= 10 and &gt; 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 &lt; rather than a &lt;= 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">&lt;&lt;</span> <span class="string">"Probability of getting less than "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span>
 178   <span class="special">&lt;&lt;</span> <span class="string">" (&lt; "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span> <span class="special">&lt;&lt;</span> <span class="string">") answers right by guessing is "</span>
 179   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 180 </pre>
 181 <pre class="programlisting">Probability of getting less than 11 (&lt; 11) answers right by guessing is 0.999715
 182 </pre>
 183 <p>
 184             and similarly to get a &gt;= rather than a &gt; 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">&lt;&lt;</span> <span class="string">"Probability of getting at least "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span>
 191   <span class="special">&lt;&lt;</span> <span class="string">"(&gt;= "</span> <span class="special">&lt;&lt;</span> <span class="identifier">pass_score</span> <span class="special">&lt;&lt;</span> <span class="string">") answers right by guessing is "</span>
 192   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">", only 1 in "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</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 (&gt;= 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">&lt;&lt;</span> <span class="string">"\n"</span> <span class="string">"At most (&lt;=)"</span><span class="string">"\n"</span><span class="string">"Guessed OK   Probability"</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</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">&lt;=</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">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">score</span> <span class="special">&lt;&lt;</span> <span class="string">"           "</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span>
 206     <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 207 <span class="special">}</span>
 208 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 209 </pre>
 210 <pre class="programlisting">At most (&lt;=)
 211 Guessed 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
 222 10           0.9997147608
 223 11           0.9999618928
 224 12           0.9999962167
 225 13           0.9999997371
 226 14           0.9999999886
 227 15           0.9999999998
 228 16           1
 229 </pre>
 230 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span> <span class="string">"At least (&gt;)"</span><span class="string">"\n"</span><span class="string">"Guessed OK   Probability"</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</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">&lt;=</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">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">score</span> <span class="special">&lt;&lt;</span> <span class="string">"           "</span>  <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span>
 234     <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 235 <span class="special">}</span>
 236 </pre>
 237 <pre class="programlisting">At least (&gt;)
 238 Guessed 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
 249 10           0.0002852391917
 250 11           3.810715862e-005
 251 12           3.783265129e-006
 252 13           2.628657967e-007
 253 14           1.140870154e-008
 254 15           2.328306437e-010
 255 16           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">&lt;=</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">&lt;&lt;</span> <span class="string">"Probability of getting between "</span>
 274   <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
 275   <span class="special">&lt;&lt;</span> <span class="identifier">sum</span>  <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
 283   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
 292   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
 295   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 &lt;= 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">&lt;&lt;</span> <span class="string">"Probability of getting between "</span> <span class="special">&lt;&lt;</span> <span class="identifier">low</span> <span class="special">&lt;&lt;</span> <span class="string">" and "</span> <span class="special">&lt;&lt;</span> <span class="identifier">high</span> <span class="special">&lt;&lt;</span> <span class="string">" answers right by guessing is "</span>
 298   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 4 &lt;= 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
 301 Probability of getting between 1 and 8 answers right by guessing is 0.9825
 302 Probability 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">&lt;&lt;</span> <span class="string">"By guessing, on average, one can expect to get "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">" correct answers."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 314 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard deviation is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 315 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span>
 316   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" and "</span>
 317   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" correct."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 318 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Mode (the most frequent) is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 319 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Skewness is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">&lt;&lt;</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.
 322 Standard deviation is 1.732
 323 So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct.
 324 Mode (the most frequent) is 4
 325 Skewness 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">&lt;&lt;</span> <span class="string">"Quartiles "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 336   <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles</span>
 337 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"1 standard deviation "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 338   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Deciles "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 340   <span class="special">&lt;&lt;</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">))&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles</span>
 341 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"5 to 95% "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 342   <span class="special">&lt;&lt;</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">))&lt;&lt;</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">&lt;&lt;</span> <span class="string">"2.5 to 97.5% "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 344   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"2 to 98% "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 346   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"If guessing then percentiles 1 to 99% will get "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" right."</span> <span class="special">&lt;&lt;</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
 356 1 standard deviation 2 to 5
 357 Deciles 1 to 6
 358 5 to 95% 0 to 7
 359 2.5 to 97.5% 0 to 8
 360 2 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">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">real</span><span class="special">&gt;</span> <span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">real_quantile_policy</span><span class="special">&gt;</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">&lt;&lt;</span> <span class="string">"Quartiles "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 413   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"1 standard deviation "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 415   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"Deciles "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 417   <span class="special">&lt;&lt;</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">))&lt;&lt;</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">&lt;&lt;</span> <span class="string">"5 to 95% "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 419   <span class="special">&lt;&lt;</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">))&lt;&lt;</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">&lt;&lt;</span> <span class="string">"2.5 to 97.5% "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 421   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"2 to 98% "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span>
 423   <span class="special">&lt;&lt;</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">&lt;&lt;</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">&lt;&lt;</span> <span class="string">"If guessing, then percentiles 1 to 99% will get "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" to "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="string">" right."</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 427 </pre>
 428 <pre class="programlisting">Real Quantiles
 429 Quartiles 2 to 4.621
 430 1 standard deviation 2.665 to 4.194
 431 Deciles 1.349 to 5.758
 432 5 to 95% 0.8374 to 6.456
 433 2.5 to 97.5% 0.4281 to 7.069
 434 2 to 98% 0.3131 to 7.252
 435 If 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 &#169; 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       R&#229;de, 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>
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