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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h5 class="title"> | |
27 | <a name="math_toolkit.stat_tut.weg.binom_eg.binomial_quiz_example"></a><a class="link" href="binomial_quiz_example.html" title="Binomial Quiz Example">Binomial | |
28 | Quiz Example</a> | |
29 | </h5></div></div></div> | |
30 | <p> | |
31 | A multiple choice test has four possible answers to each of 16 questions. | |
32 | A student guesses the answer to each question, so the probability of | |
33 | getting a correct answer on any given question is one in four, a quarter, | |
34 | 1/4, 25% or fraction 0.25. The conditions of the binomial experiment | |
35 | are assumed to be met: n = 16 questions constitute the trials; each question | |
36 | results in one of two possible outcomes (correct or incorrect); the probability | |
37 | of being correct is 0.25 and is constant if no knowledge about the subject | |
38 | is assumed; the questions are answered independently if the student's | |
39 | answer to a question in no way influences his/her answer to another question. | |
40 | </p> | |
41 | <p> | |
42 | First, we need to be able to use the binomial distribution constructor | |
43 | (and some std input/output, of course). | |
44 | </p> | |
45 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
46 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial</span><span class="special">;</span> | |
47 | ||
48 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> | |
49 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
50 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ios</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">flush</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> | |
51 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> | |
52 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> | |
53 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">exception</span><span class="special">></span> | |
54 | </pre> | |
55 | <p> | |
56 | The number of correct answers, X, is distributed as a binomial random | |
57 | variable with binomial distribution parameters: questions n and success | |
58 | fraction probability p. So we construct a binomial distribution: | |
59 | </p> | |
60 | <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">questions</span> <span class="special">=</span> <span class="number">16</span><span class="special">;</span> <span class="comment">// All the questions in the quiz.</span> | |
61 | <span class="keyword">int</span> <span class="identifier">answers</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="comment">// Possible answers to each question.</span> | |
62 | <span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">1.</span> <span class="special">/</span> <span class="identifier">answers</span><span class="special">;</span> <span class="comment">// If a random guess, p = 1/4 = 0.25.</span> | |
63 | <span class="identifier">binomial</span> <span class="identifier">quiz</span><span class="special">(</span><span class="identifier">questions</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> | |
64 | </pre> | |
65 | <p> | |
66 | and display the distribution parameters we used thus: | |
67 | </p> | |
68 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"In a quiz with "</span> <span class="special"><<</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">trials</span><span class="special">()</span> | |
69 | <span class="special"><<</span> <span class="string">" questions and with a probability of guessing right of "</span> | |
70 | <span class="special"><<</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special">*</span> <span class="number">100</span> <span class="special"><<</span> <span class="string">" %"</span> | |
71 | <span class="special"><<</span> <span class="string">" or 1 in "</span> <span class="special"><<</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="number">1.</span> <span class="special">/</span> <span class="identifier">quiz</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">())</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
72 | </pre> | |
73 | <p> | |
74 | Show a few probabilities of just guessing: | |
75 | </p> | |
76 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting none right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.010023</span> | |
77 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly one right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
78 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly two right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
79 | <span class="keyword">int</span> <span class="identifier">pass_score</span> <span class="special">=</span> <span class="number">11</span><span class="special">;</span> | |
80 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting exactly "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> <span class="special"><<</span> <span class="string">" answers right by chance is "</span> | |
81 | <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
82 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting all "</span> <span class="special"><<</span> <span class="identifier">questions</span> <span class="special"><<</span> <span class="string">" answers right by chance is "</span> | |
83 | <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">questions</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
84 | </pre> | |
85 | <pre class="programlisting">Probability of getting none right is 0.0100226 | |
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"><<</span> <span class="string">"\n"</span> <span class="string">"Guessed Probability"</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
98 | <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">successes</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">successes</span> <span class="special"><=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">successes</span><span class="special">++)</span> | |
99 | <span class="special">{</span> | |
100 | <span class="keyword">double</span> <span class="identifier">probability</span> <span class="special">=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">successes</span><span class="special">);</span> | |
101 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">successes</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">probability</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
102 | <span class="special">}</span> | |
103 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
104 | </pre> | |
105 | <pre class="programlisting">Guessed Probability | |
106 | 0 0.0100226 | |
107 | 1 0.0534538 | |
108 | 2 0.133635 | |
109 | 3 0.207876 | |
110 | 4 0.225199 | |
111 | 5 0.180159 | |
112 | 6 0.110097 | |
113 | 7 0.0524273 | |
114 | 8 0.0196602 | |
115 | 9 0.00582526 | |
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"><<</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
128 | </pre> | |
129 | <pre class="programlisting">Probability of getting none or one right is 0.0634764 | |
130 | </pre> | |
131 | <p> | |
132 | But if more than a couple of scores are involved, it is more convenient | |
133 | (and may be more accurate) to use the Cumulative Distribution Function | |
134 | (cdf) instead: | |
135 | </p> | |
136 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting none or one right is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
137 | </pre> | |
138 | <pre class="programlisting">Probability of getting none or one right is 0.0634764 | |
139 | </pre> | |
140 | <p> | |
141 | Since the cdf is inclusive, we can get the probability of getting up | |
142 | to 10 right ( <= ) | |
143 | </p> | |
144 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting <= 10 right (to fail) is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
145 | </pre> | |
146 | <pre class="programlisting">Probability of getting <= 10 right (to fail) is 0.999715 | |
147 | </pre> | |
148 | <p> | |
149 | To get the probability of getting 11 or more right (to pass), it is tempting | |
150 | to use | |
151 | </p> | |
152 | <pre class="programlisting"><span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span></pre> | |
153 | <p> | |
154 | to get the probability of > 10 | |
155 | </p> | |
156 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting > 10 right (to pass) is "</span> <span class="special"><<</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
157 | </pre> | |
158 | <pre class="programlisting">Probability of getting > 10 right (to pass) is 0.000285239 | |
159 | </pre> | |
160 | <p> | |
161 | But this should be resisted in favor of using the <a class="link" href="../../overview/complements.html" title="Complements are supported too - and when to use them">complements</a> | |
162 | function (see <a class="link" href="../../overview/complements.html#why_complements">why complements?</a>). | |
163 | </p> | |
164 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting > 10 right (to pass) is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
165 | </pre> | |
166 | <pre class="programlisting">Probability of getting > 10 right (to pass) is 0.000285239 | |
167 | </pre> | |
168 | <p> | |
169 | And we can check that these two, <= 10 and > 10, add up to unity. | |
170 | </p> | |
171 | <pre class="programlisting"><span class="identifier">BOOST_ASSERT</span><span class="special">((</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">10</span><span class="special">)))</span> <span class="special">==</span> <span class="number">1.</span><span class="special">);</span> | |
172 | </pre> | |
173 | <p> | |
174 | If we want a < rather than a <= test, because the CDF is inclusive, | |
175 | we must subtract one from the score. | |
176 | </p> | |
177 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting less than "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> | |
178 | <span class="special"><<</span> <span class="string">" (< "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> <span class="special"><<</span> <span class="string">") answers right by guessing is "</span> | |
179 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
180 | </pre> | |
181 | <pre class="programlisting">Probability of getting less than 11 (< 11) answers right by guessing is 0.999715 | |
182 | </pre> | |
183 | <p> | |
184 | and similarly to get a >= rather than a > test we also need to | |
185 | subtract one from the score (and can again check the sum is unity). This | |
186 | is because if the cdf is <span class="emphasis"><em>inclusive</em></span>, then its complement | |
187 | must be <span class="emphasis"><em>exclusive</em></span> otherwise there would be one possible | |
188 | outcome counted twice! | |
189 | </p> | |
190 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting at least "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> | |
191 | <span class="special"><<</span> <span class="string">"(>= "</span> <span class="special"><<</span> <span class="identifier">pass_score</span> <span class="special"><<</span> <span class="string">") answers right by guessing is "</span> | |
192 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">))</span> | |
193 | <span class="special"><<</span> <span class="string">", only 1 in "</span> <span class="special"><<</span> <span class="number">1</span><span class="special">/</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
194 | ||
195 | <span class="identifier">BOOST_ASSERT</span><span class="special">((</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">pass_score</span><span class="special">-</span><span class="number">1</span><span class="special">)))</span> <span class="special">==</span> <span class="number">1</span><span class="special">);</span> | |
196 | </pre> | |
197 | <pre class="programlisting">Probability of getting at least 11 (>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83 | |
198 | </pre> | |
199 | <p> | |
200 | Finally we can tabulate some probabilities: | |
201 | </p> | |
202 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="string">"At most (<=)"</span><span class="string">"\n"</span><span class="string">"Guessed OK Probability"</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
203 | <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">score</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">score</span> <span class="special"><=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">score</span><span class="special">++)</span> | |
204 | <span class="special">{</span> | |
205 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">score</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> | |
206 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">score</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
207 | <span class="special">}</span> | |
208 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
209 | </pre> | |
210 | <pre class="programlisting">At most (<=) | |
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"><<</span> <span class="string">"\n"</span> <span class="string">"At least (>)"</span><span class="string">"\n"</span><span class="string">"Guessed OK Probability"</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
231 | <span class="keyword">for</span> <span class="special">(</span><span class="keyword">int</span> <span class="identifier">score</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">score</span> <span class="special"><=</span> <span class="identifier">questions</span><span class="special">;</span> <span class="identifier">score</span><span class="special">++)</span> | |
232 | <span class="special">{</span> | |
233 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">score</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> | |
234 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">score</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
235 | <span class="special">}</span> | |
236 | </pre> | |
237 | <pre class="programlisting">At least (>) | |
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"><=</span> <span class="identifier">high</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++)</span> | |
269 | <span class="special">{</span> | |
270 | <span class="identifier">sum</span> <span class="special">+=</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">i</span><span class="special">);</span> | |
271 | <span class="special">}</span> | |
272 | <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">4</span><span class="special">);</span> | |
273 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> | |
274 | <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> | |
275 | <span class="special"><<</span> <span class="identifier">sum</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.61323</span> | |
276 | </pre> | |
277 | <pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132 | |
278 | </pre> | |
279 | <p> | |
280 | Or, usually better, we can use the difference of cdfs instead: | |
281 | </p> | |
282 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> | |
283 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.61323</span> | |
284 | </pre> | |
285 | <pre class="programlisting">Probability of getting between 3 and 5 answers right by guessing is 0.6132 | |
286 | </pre> | |
287 | <p> | |
288 | And we can also try a few more combinations of high and low choices: | |
289 | </p> | |
290 | <pre class="programlisting"><span class="identifier">low</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span> | |
291 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> | |
292 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 and 6 P= 0.91042</span> | |
293 | <span class="identifier">low</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">8</span><span class="special">;</span> | |
294 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> | |
295 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 <= x 8 P = 0.9825</span> | |
296 | <span class="identifier">low</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> <span class="identifier">high</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span> | |
297 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of getting between "</span> <span class="special"><<</span> <span class="identifier">low</span> <span class="special"><<</span> <span class="string">" and "</span> <span class="special"><<</span> <span class="identifier">high</span> <span class="special"><<</span> <span class="string">" answers right by guessing is "</span> | |
298 | <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">high</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="identifier">low</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 4 <= x 4 P = 0.22520</span> | |
299 | </pre> | |
300 | <pre class="programlisting">Probability of getting between 1 and 6 answers right by guessing is 0.9104 | |
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"><<</span> <span class="string">"By guessing, on average, one can expect to get "</span> <span class="special"><<</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" correct answers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
314 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviation is "</span> <span class="special"><<</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
315 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"So about 2/3 will lie within 1 standard deviation and get between "</span> | |
316 | <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" and "</span> | |
317 | <span class="special"><<</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">mean</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" correct."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
318 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Mode (the most frequent) is "</span> <span class="special"><<</span> <span class="identifier">mode</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
319 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Skewness is "</span> <span class="special"><<</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
320 | </pre> | |
321 | <pre class="programlisting">By guessing, on average, one can expect to get 4 correct answers. | |
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"><<</span> <span class="string">"Quartiles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
336 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles</span> | |
337 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"1 standard deviation "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.33</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
338 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.67</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 sd</span> | |
339 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Deciles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.1</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
340 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.1</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles</span> | |
341 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"5 to 95% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
342 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5 to 95%</span> | |
343 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2.5 to 97.5% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
344 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.025</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.5 to 97.5%</span> | |
345 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2 to 98% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.02</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
346 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.02</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2 to 98%</span> | |
347 | ||
348 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If guessing then percentiles 1 to 99% will get "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> | |
349 | <span class="special"><<</span> <span class="string">" to "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" right."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
350 | </pre> | |
351 | <p> | |
352 | Notice that these output integral values because the default policy is | |
353 | <code class="computeroutput"><span class="identifier">integer_round_outwards</span></code>. | |
354 | </p> | |
355 | <pre class="programlisting">Quartiles 2 to 5 | |
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"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">real</span><span class="special">></span> <span class="special">></span> <span class="identifier">real_quantile_policy</span><span class="special">;</span> | |
391 | </pre> | |
392 | <p> | |
393 | Add a custom binomial distribution called | |
394 | </p> | |
395 | <pre class="programlisting"><span class="identifier">real_quantile_binomial</span></pre> | |
396 | <p> | |
397 | that uses | |
398 | </p> | |
399 | <pre class="programlisting"><span class="identifier">real_quantile_policy</span></pre> | |
400 | <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">binomial_distribution</span><span class="special">;</span> | |
401 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">real_quantile_policy</span><span class="special">></span> <span class="identifier">real_quantile_binomial</span><span class="special">;</span> | |
402 | </pre> | |
403 | <p> | |
404 | Construct an object of this custom distribution: | |
405 | </p> | |
406 | <pre class="programlisting"><span class="identifier">real_quantile_binomial</span> <span class="identifier">quiz_real</span><span class="special">(</span><span class="identifier">questions</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> | |
407 | </pre> | |
408 | <p> | |
409 | And use this to show some quantiles - that now have real rather than | |
410 | integer values. | |
411 | </p> | |
412 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quartiles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz</span><span class="special">,</span> <span class="number">0.25</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
413 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.25</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Quartiles 2 to 4.6212</span> | |
414 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"1 standard deviation "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.33</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
415 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.67</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1 sd 2.6654 4.194</span> | |
416 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Deciles "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.1</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
417 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.1</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Deciles 1.3487 5.7583</span> | |
418 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"5 to 95% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
419 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))<<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5 to 95% 0.83739 6.4559</span> | |
420 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2.5 to 97.5% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.025</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
421 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.025</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.5 to 97.5% 0.42806 7.0688</span> | |
422 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"2 to 98% "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.02</span><span class="special">)</span> <span class="special"><<</span> <span class="string">" to "</span> | |
423 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.02</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2 to 98% 0.31311 7.7880</span> | |
424 | ||
425 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"If guessing, then percentiles 1 to 99% will get "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> | |
426 | <span class="special"><<</span> <span class="string">" to "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">quiz_real</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" right."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
427 | </pre> | |
428 | <pre class="programlisting">Real Quantiles | |
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 © 2006-2010, 2012-2014 Nikhar Agrawal, | |
445 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
446 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
447 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
448 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
449 | 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>) | |
450 | </p> | |
451 | </div></td> | |
452 | </tr></table> | |
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