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  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">&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   <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">;</span>
  55 </pre>
  56 <p>
  57             The number of correct answers, X, is distributed as a binomial random
  58             variable with binomial distribution parameters: questions n and success
  59             fraction probability p. So we construct a binomial distribution:
  60           </p>
  61 <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>
  62 <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>
  63 <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>
  64 <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>
  65 </pre>
  66 <p>
  67             and display the distribution parameters we used thus:
  68           </p>
  69 <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>
  70   <span class="special">&lt;&lt;</span> <span class="string">" questions and with a probability of guessing right of "</span>
  71   <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>
  72   <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>
  73 </pre>
  74 <p>
  75             Show a few probabilities of just guessing:
  76           </p>
  77 <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>
  78 <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>
  79 <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>
  80 <span class="keyword">int</span> <span class="identifier">pass_score</span> <span class="special">=</span> <span class="number">11</span><span class="special">;</span>
  81 <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>
  82   <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>
  83 <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>
  84   <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>
  85 </pre>
  86 <pre class="programlisting">Probability of getting none right is 0.0100226
  87 Probability of getting exactly one right is 0.0534538
  88 Probability of getting exactly two right is 0.133635
  89 Probability of getting exactly 11 right is 0.000247132
  90 Probability of getting exactly all 16 answers right by chance is 2.32831e-010
  91 </pre>
  92 <p>
  93             These don't give any encouragement to guessers!
  94           </p>
  95 <p>
  96             We can tabulate the 'getting exactly right' ( == ) probabilities thus:
  97           </p>
  98 <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>
  99 <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>
 100 <span class="special">{</span>
 101   <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>
 102   <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>
 103 <span class="special">}</span>
 104 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 105 </pre>
 106 <pre class="programlisting">Guessed Probability
 107  0      0.0100226
 108  1      0.0534538
 109  2      0.133635
 110  3      0.207876
 111  4      0.225199
 112  5      0.180159
 113  6      0.110097
 114  7      0.0524273
 115  8      0.0196602
 116  9      0.00582526
 117 10      0.00135923
 118 11      0.000247132
 119 12      3.43239e-005
 120 13      3.5204e-006
 121 14      2.51457e-007
 122 15      1.11759e-008
 123 16      2.32831e-010
 124 </pre>
 125 <p>
 126             Then we can add the probabilities of some 'exactly right' like this:
 127           </p>
 128 <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>
 129 </pre>
 130 <pre class="programlisting">Probability of getting none or one right is 0.0634764
 131 </pre>
 132 <p>
 133             But if more than a couple of scores are involved, it is more convenient
 134             (and may be more accurate) to use the Cumulative Distribution Function
 135             (cdf) instead:
 136           </p>
 137 <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>
 138 </pre>
 139 <pre class="programlisting">Probability of getting none or one right is 0.0634764
 140 </pre>
 141 <p>
 142             Since the cdf is inclusive, we can get the probability of getting up
 143             to 10 right ( &lt;= )
 144           </p>
 145 <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>
 146 </pre>
 147 <pre class="programlisting">Probability of getting &lt;= 10 right (to fail) is 0.999715
 148 </pre>
 149 <p>
 150             To get the probability of getting 11 or more right (to pass), it is tempting
 151             to use
 152           </p>
 153 <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>
 154 <p>
 155             to get the probability of &gt; 10
 156           </p>
 157 <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>
 158 </pre>
 159 <pre class="programlisting">Probability of getting &gt; 10 right (to pass) is 0.000285239
 160 </pre>
 161 <p>
 162             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>
 163             function (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>).
 164           </p>
 165 <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>
 166 </pre>
 167 <pre class="programlisting">Probability of getting &gt; 10 right (to pass) is 0.000285239
 168 </pre>
 169 <p>
 170             And we can check that these two, &lt;= 10 and &gt; 10, add up to unity.
 171           </p>
 172 <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>
 173 </pre>
 174 <p>
 175             If we want a &lt; rather than a &lt;= test, because the CDF is inclusive,
 176             we must subtract one from the score.
 177           </p>
 178 <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>
 179   <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>
 180   <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>
 181 </pre>
 182 <pre class="programlisting">Probability of getting less than 11 (&lt; 11) answers right by guessing is 0.999715
 183 </pre>
 184 <p>
 185             and similarly to get a &gt;= rather than a &gt; test we also need to
 186             subtract one from the score (and can again check the sum is unity). This
 187             is because if the cdf is <span class="emphasis"><em>inclusive</em></span>, then its complement
 188             must be <span class="emphasis"><em>exclusive</em></span> otherwise there would be one possible
 189             outcome counted twice!
 190           </p>
 191 <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>
 192   <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>
 193   <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>
 194   <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>
 195
 196 <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>
 197 </pre>
 198 <pre class="programlisting">Probability of getting at least 11 (&gt;= 11) answers right by guessing is 0.000285239, only 1 in 3505.83
 199 </pre>
 200 <p>
 201             Finally we can tabulate some probabilities:
 202           </p>
 203 <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>
 204 <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>
 205 <span class="special">{</span>
 206   <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>
 207     <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>
 208 <span class="special">}</span>
 209 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 210 </pre>
 211 <pre class="programlisting">At most (&lt;=)
 212 Guessed OK   Probability
 213  0           0.01002259576
 214  1           0.0634764398
 215  2           0.1971110499
 216  3           0.4049871101
 217  4           0.6301861752
 218  5           0.8103454274
 219  6           0.9204427481
 220  7           0.9728700437
 221  8           0.9925302796
 222  9           0.9983555346
 223 10           0.9997147608
 224 11           0.9999618928
 225 12           0.9999962167
 226 13           0.9999997371
 227 14           0.9999999886
 228 15           0.9999999998
 229 16           1
 230 </pre>
 231 <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>
 232 <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>
 233 <span class="special">{</span>
 234   <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>
 235     <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>
 236 <span class="special">}</span>
 237 </pre>
 238 <pre class="programlisting">At least (&gt;)
 239 Guessed OK   Probability
 240  0           0.9899774042
 241  1           0.9365235602
 242  2           0.8028889501
 243  3           0.5950128899
 244  4           0.3698138248
 245  5           0.1896545726
 246  6           0.07955725188
 247  7           0.02712995629
 248  8           0.00746972044
 249  9           0.001644465374
 250 10           0.0002852391917
 251 11           3.810715862e-005
 252 12           3.783265129e-006
 253 13           2.628657967e-007
 254 14           1.140870154e-008
 255 15           2.328306437e-010
 256 16           0
 257 </pre>
 258 <p>
 259             We now consider the probabilities of <span class="bold"><strong>ranges</strong></span>
 260             of correct guesses.
 261           </p>
 262 <p>
 263             First, calculate the probability of getting a range of guesses right,
 264             by adding the exact probabilities of each from low ... high.
 265           </p>
 266 <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>
 267 <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>
 268 <span class="keyword">double</span> <span class="identifier">sum</span> <span class="special">=</span> <span class="number">0.</span><span class="special">;</span>
 269 <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>
 270 <span class="special">{</span>
 271   <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>
 272 <span class="special">}</span>
 273 <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>
 274 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Probability of getting between "</span>
 275   <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>
 276   <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>
 277 </pre>
 278 <pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132
 279 </pre>
 280 <p>
 281             Or, usually better, we can use the difference of cdfs instead:
 282           </p>
 283 <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>
 284   <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>
 285 </pre>
 286 <pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132
 287 </pre>
 288 <p>
 289             And we can also try a few more combinations of high and low choices:
 290           </p>
 291 <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>
 292 <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>
 293   <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>
 294 <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>
 295 <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>
 296   <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>
 297 <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>
 298 <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>
 299   <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>
 300 </pre>
 301 <pre class="programlisting">Probability of getting between 1 and 6 answers right by guessing is 0.9104
 302 Probability of getting between 1 and 8 answers right by guessing is 0.9825
 303 Probability of getting between 4 and 4 answers right by guessing is 0.2252
 304 </pre>
 305 <h5>
 306 <a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.h0"></a>
 307             <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
 308             Binomial distribution moments</a>
 309           </h5>
 310 <p>
 311             Using moments of the distribution, we can say more about the spread of
 312             results from guessing.
 313           </p>
 314 <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>
 315 <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>
 316 <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>
 317   <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>
 318   <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>
 319 <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>
 320 <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>
 321 </pre>
 322 <pre class="programlisting">By guessing, on average, one can expect to get 4 correct answers.
 323 Standard deviation is 1.732
 324 So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct.
 325 Mode (the most frequent) is 4
 326 Skewness is 0.2887
 327 </pre>
 328 <h5>
 329 <a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example.h1"></a>
 330             <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>
 331           </h5>
 332 <p>
 333             The quantiles (percentiles or percentage points) for a few probability
 334             levels:
 335           </p>
 336 <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>
 337   <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>
 338 <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>
 339   <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>
 340 <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>
 341   <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>
 342 <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>
 343   <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>
 344 <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>
 345   <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>
 346 <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>
 347   <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>
 348
 349 <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>
 350   <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>
 351 </pre>
 352 <p>
 353             Notice that these output integral values because the default policy is
 354             <code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>.
 355           </p>
 356 <pre class="programlisting">Quartiles 2 to 5
 357 1 standard deviation 2 to 5
 358 Deciles 1 to 6
 359 5 to 95% 0 to 7
 360 2.5 to 97.5% 0 to 8
 361 2 to 98% 0 to 8
 362 </pre>
 363 <p>
 364             Quantiles values are controlled by the <a class="link" href="../../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">understanding
 365             discrete quantiles</a> quantile policy chosen. The default is <code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>, so the lower
 366             quantile is rounded down, and the upper quantile is rounded up.
 367           </p>
 368 <p>
 369             But we might believe that the real values tell us a little more - see
 370             <a class="link" href="../../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">discrete functions</a>.
 371           </p>
 372 <p>
 373             We could control the policy for <span class="bold"><strong>all</strong></span>
 374             distributions by
 375           </p>
 376 <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span>
 377
 378 <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>
 379 </pre>
 380 <p>
 381             to this <span class="bold"><strong>one, and only</strong></span>, translation unit.
 382           </p>
 383 <p>
 384             Or we can now create a (typedef for) policy that has discrete quantiles
 385             real (here avoiding any 'using namespaces ...' statements):
 386           </p>
 387 <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>
 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">discrete_quantile</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">real</span><span class="special">;</span>
 390 <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>
 391 <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>
 392 </pre>
 393 <p>
 394             Add a custom binomial distribution called
 395           </p>
 396 <pre class="programlisting"><span class="identifier">real_quantile_binomial</span></pre>
 397 <p>
 398             that uses
 399           </p>
 400 <pre class="programlisting"><span class="identifier">real_quantile_policy</span></pre>
 401 <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>
 402 <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>
 403 </pre>
 404 <p>
 405             Construct an object of this custom distribution:
 406           </p>
 407 <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>
 408 </pre>
 409 <p>
 410             And use this to show some quantiles - that now have real rather than
 411             integer values.
 412           </p>
 413 <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>
 414   <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>
 415 <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>
 416   <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>
 417 <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>
 418   <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>
 419 <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>
 420   <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>
 421 <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>
 422   <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>
 423 <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>
 424   <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>
 425
 426 <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>
 427   <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>
 428 </pre>
 429 <pre class="programlisting">Real Quantiles
 430 Quartiles 2 to 4.621
 431 1 standard deviation 2.665 to 4.194
 432 Deciles 1.349 to 5.758
 433 5 to 95% 0.8374 to 6.456
 434 2.5 to 97.5% 0.4281 to 7.069
 435 2 to 98% 0.3131 to 7.252
 436 If guessing then percentiles 1 to 99% will get 0 to 7.788 right.
 437 </pre>
 438 <p>
 439             See <a href="../../../../../../example/binomial_quiz_example.cpp" target="_top">binomial_quiz_example.cpp</a>
 440             for full source code and output.
 441           </p>
 442 </div>
 443 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 444 <td align="left"></td>
 445 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 446       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 447       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 448       Thijs van den Berg, 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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