summaryrefslogtreecommitdiffstats
path: root/src/boost/libs/math/example/binomial_quiz_example.cpp
diff options
context:
space:
mode:
Diffstat (limited to 'src/boost/libs/math/example/binomial_quiz_example.cpp')
-rw-r--r--src/boost/libs/math/example/binomial_quiz_example.cpp525
1 files changed, 525 insertions, 0 deletions
diff --git a/src/boost/libs/math/example/binomial_quiz_example.cpp b/src/boost/libs/math/example/binomial_quiz_example.cpp
new file mode 100644
index 00000000..acf06e2d
--- /dev/null
+++ b/src/boost/libs/math/example/binomial_quiz_example.cpp
@@ -0,0 +1,525 @@
+// Copyright Paul A. Bristow 2007, 2009, 2010
+// Copyright John Maddock 2006
+
+// Use, modification and distribution are subject to the
+// Boost Software License, Version 1.0.
+// (See accompanying file LICENSE_1_0.txt
+// or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+// binomial_examples_quiz.cpp
+
+// Simple example of computing probabilities and quantiles for a binomial random variable
+// representing the correct guesses on a multiple-choice test.
+
+// source http://www.stat.wvu.edu/SRS/Modules/Binomial/test.html
+
+//[binomial_quiz_example1
+/*`
+A multiple choice test has four possible answers to each of 16 questions.
+A student guesses the answer to each question,
+so the probability of getting a correct answer on any given question is
+one in four, a quarter, 1/4, 25% or fraction 0.25.
+The conditions of the binomial experiment are assumed to be met:
+n = 16 questions constitute the trials;
+each question results in one of two possible outcomes (correct or incorrect);
+the probability of being correct is 0.25 and is constant if no knowledge about the subject is assumed;
+the questions are answered independently if the student's answer to a question
+in no way influences his/her answer to another question.
+
+First, we need to be able to use the binomial distribution constructor
+(and some std input/output, of course).
+*/
+
+#include <boost/math/distributions/binomial.hpp>
+ using boost::math::binomial;
+
+#include <iostream>
+ using std::cout; using std::endl;
+ using std::ios; using std::flush; using std::left; using std::right; using std::fixed;
+#include <iomanip>
+ using std::setw; using std::setprecision;
+#include <exception>
+
+
+
+//][/binomial_quiz_example1]
+
+int main()
+{
+ try
+ {
+ cout << "Binomial distribution example - guessing in a quiz." << endl;
+//[binomial_quiz_example2
+/*`
+The number of correct answers, X, is distributed as a binomial random variable
+with binomial distribution parameters: questions n and success fraction probability p.
+So we construct a binomial distribution:
+*/
+ int questions = 16; // All the questions in the quiz.
+ int answers = 4; // Possible answers to each question.
+ double success_fraction = 1. / answers; // If a random guess, p = 1/4 = 0.25.
+ binomial quiz(questions, success_fraction);
+/*`
+and display the distribution parameters we used thus:
+*/
+ cout << "In a quiz with " << quiz.trials()
+ << " questions and with a probability of guessing right of "
+ << quiz.success_fraction() * 100 << " %"
+ << " or 1 in " << static_cast<int>(1. / quiz.success_fraction()) << endl;
+/*`
+Show a few probabilities of just guessing:
+*/
+ cout << "Probability of getting none right is " << pdf(quiz, 0) << endl; // 0.010023
+ cout << "Probability of getting exactly one right is " << pdf(quiz, 1) << endl;
+ cout << "Probability of getting exactly two right is " << pdf(quiz, 2) << endl;
+ int pass_score = 11;
+ cout << "Probability of getting exactly " << pass_score << " answers right by chance is "
+ << pdf(quiz, pass_score) << endl;
+ cout << "Probability of getting all " << questions << " answers right by chance is "
+ << pdf(quiz, questions) << endl;
+/*`
+[pre
+Probability of getting none right is 0.0100226
+Probability of getting exactly one right is 0.0534538
+Probability of getting exactly two right is 0.133635
+Probability of getting exactly 11 right is 0.000247132
+Probability of getting exactly all 16 answers right by chance is 2.32831e-010
+]
+These don't give any encouragement to guessers!
+
+We can tabulate the 'getting exactly right' ( == ) probabilities thus:
+*/
+ cout << "\n" "Guessed Probability" << right << endl;
+ for (int successes = 0; successes <= questions; successes++)
+ {
+ double probability = pdf(quiz, successes);
+ cout << setw(2) << successes << " " << probability << endl;
+ }
+ cout << endl;
+/*`
+[pre
+Guessed Probability
+ 0 0.0100226
+ 1 0.0534538
+ 2 0.133635
+ 3 0.207876
+ 4 0.225199
+ 5 0.180159
+ 6 0.110097
+ 7 0.0524273
+ 8 0.0196602
+ 9 0.00582526
+10 0.00135923
+11 0.000247132
+12 3.43239e-005
+13 3.5204e-006
+14 2.51457e-007
+15 1.11759e-008
+16 2.32831e-010
+]
+Then we can add the probabilities of some 'exactly right' like this:
+*/
+ cout << "Probability of getting none or one right is " << pdf(quiz, 0) + pdf(quiz, 1) << endl;
+
+/*`
+[pre
+Probability of getting none or one right is 0.0634764
+]
+But if more than a couple of scores are involved, it is more convenient (and may be more accurate)
+to use the Cumulative Distribution Function (cdf) instead:
+*/
+ cout << "Probability of getting none or one right is " << cdf(quiz, 1) << endl;
+/*`
+[pre
+Probability of getting none or one right is 0.0634764
+]
+Since the cdf is inclusive, we can get the probability of getting up to 10 right ( <= )
+*/
+ cout << "Probability of getting <= 10 right (to fail) is " << cdf(quiz, 10) << endl;
+/*`
+[pre
+Probability of getting <= 10 right (to fail) is 0.999715
+]
+To get the probability of getting 11 or more right (to pass),
+it is tempting to use ``1 - cdf(quiz, 10)`` to get the probability of > 10
+*/
+ cout << "Probability of getting > 10 right (to pass) is " << 1 - cdf(quiz, 10) << endl;
+/*`
+[pre
+Probability of getting > 10 right (to pass) is 0.000285239
+]
+But this should be resisted in favor of using the __complements function (see __why_complements).
+*/
+ cout << "Probability of getting > 10 right (to pass) is " << cdf(complement(quiz, 10)) << endl;
+/*`
+[pre
+Probability of getting > 10 right (to pass) is 0.000285239
+]
+And we can check that these two, <= 10 and > 10, add up to unity.
+*/
+BOOST_ASSERT((cdf(quiz, 10) + cdf(complement(quiz, 10))) == 1.);
+/*`
+If we want a < rather than a <= test, because the CDF is inclusive, we must subtract one from the score.
+*/
+ cout << "Probability of getting less than " << pass_score
+ << " (< " << pass_score << ") answers right by guessing is "
+ << cdf(quiz, pass_score -1) << endl;
+/*`
+[pre
+Probability of getting less than 11 (< 11) answers right by guessing is 0.999715
+]
+and similarly to get a >= rather than a > test
+we also need to subtract one from the score (and can again check the sum is unity).
+This is because if the cdf is /inclusive/,
+then its complement must be /exclusive/ otherwise there would be one possible
+outcome counted twice!
+*/
+ cout << "Probability of getting at least " << pass_score
+ << "(>= " << pass_score << ") answers right by guessing is "
+ << cdf(complement(quiz, pass_score-1))
+ << ", only 1 in " << 1/cdf(complement(quiz, pass_score-1)) << endl;
+
+ BOOST_ASSERT((cdf(quiz, pass_score -1) + cdf(complement(quiz, pass_score-1))) == 1);
+
+/*`
+[pre
+Probability of getting at least 11 (>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83
+]
+Finally we can tabulate some probabilities:
+*/
+ cout << "\n" "At most (<=)""\n""Guessed OK Probability" << right << endl;
+ for (int score = 0; score <= questions; score++)
+ {
+ cout << setw(2) << score << " " << setprecision(10)
+ << cdf(quiz, score) << endl;
+ }
+ cout << endl;
+/*`
+[pre
+At most (<=)
+Guessed OK Probability
+ 0 0.01002259576
+ 1 0.0634764398
+ 2 0.1971110499
+ 3 0.4049871101
+ 4 0.6301861752
+ 5 0.8103454274
+ 6 0.9204427481
+ 7 0.9728700437
+ 8 0.9925302796
+ 9 0.9983555346
+10 0.9997147608
+11 0.9999618928
+12 0.9999962167
+13 0.9999997371
+14 0.9999999886
+15 0.9999999998
+16 1
+]
+*/
+ cout << "\n" "At least (>)""\n""Guessed OK Probability" << right << endl;
+ for (int score = 0; score <= questions; score++)
+ {
+ cout << setw(2) << score << " " << setprecision(10)
+ << cdf(complement(quiz, score)) << endl;
+ }
+/*`
+[pre
+At least (>)
+Guessed OK Probability
+ 0 0.9899774042
+ 1 0.9365235602
+ 2 0.8028889501
+ 3 0.5950128899
+ 4 0.3698138248
+ 5 0.1896545726
+ 6 0.07955725188
+ 7 0.02712995629
+ 8 0.00746972044
+ 9 0.001644465374
+10 0.0002852391917
+11 3.810715862e-005
+12 3.783265129e-006
+13 2.628657967e-007
+14 1.140870154e-008
+15 2.328306437e-010
+16 0
+]
+We now consider the probabilities of *ranges* of correct guesses.
+
+First, calculate the probability of getting a range of guesses right,
+by adding the exact probabilities of each from low ... high.
+*/
+ int low = 3; // Getting at least 3 right.
+ int high = 5; // Getting as most 5 right.
+ double sum = 0.;
+ for (int i = low; i <= high; i++)
+ {
+ sum += pdf(quiz, i);
+ }
+ cout.precision(4);
+ cout << "Probability of getting between "
+ << low << " and " << high << " answers right by guessing is "
+ << sum << endl; // 0.61323
+/*`
+[pre
+Probability of getting between 3 and 5 answers right by guessing is 0.6132
+]
+Or, usually better, we can use the difference of cdfs instead:
+*/
+ cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+ << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 0.61323
+/*`
+[pre
+Probability of getting between 3 and 5 answers right by guessing is 0.6132
+]
+And we can also try a few more combinations of high and low choices:
+*/
+ low = 1; high = 6;
+ cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+ << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 1 and 6 P= 0.91042
+ low = 1; high = 8;
+ cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+ << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 1 <= x 8 P = 0.9825
+ low = 4; high = 4;
+ cout << "Probability of getting between " << low << " and " << high << " answers right by guessing is "
+ << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 4 <= x 4 P = 0.22520
+
+/*`
+[pre
+Probability of getting between 1 and 6 answers right by guessing is 0.9104
+Probability of getting between 1 and 8 answers right by guessing is 0.9825
+Probability of getting between 4 and 4 answers right by guessing is 0.2252
+]
+[h4 Using Binomial distribution moments]
+Using moments of the distribution, we can say more about the spread of results from guessing.
+*/
+ cout << "By guessing, on average, one can expect to get " << mean(quiz) << " correct answers." << endl;
+ cout << "Standard deviation is " << standard_deviation(quiz) << endl;
+ cout << "So about 2/3 will lie within 1 standard deviation and get between "
+ << ceil(mean(quiz) - standard_deviation(quiz)) << " and "
+ << floor(mean(quiz) + standard_deviation(quiz)) << " correct." << endl;
+ cout << "Mode (the most frequent) is " << mode(quiz) << endl;
+ cout << "Skewness is " << skewness(quiz) << endl;
+
+/*`
+[pre
+By guessing, on average, one can expect to get 4 correct answers.
+Standard deviation is 1.732
+So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct.
+Mode (the most frequent) is 4
+Skewness is 0.2887
+]
+[h4 Quantiles]
+The quantiles (percentiles or percentage points) for a few probability levels:
+*/
+ cout << "Quartiles " << quantile(quiz, 0.25) << " to "
+ << quantile(complement(quiz, 0.25)) << endl; // Quartiles
+ cout << "1 standard deviation " << quantile(quiz, 0.33) << " to "
+ << quantile(quiz, 0.67) << endl; // 1 sd
+ cout << "Deciles " << quantile(quiz, 0.1) << " to "
+ << quantile(complement(quiz, 0.1))<< endl; // Deciles
+ cout << "5 to 95% " << quantile(quiz, 0.05) << " to "
+ << quantile(complement(quiz, 0.05))<< endl; // 5 to 95%
+ cout << "2.5 to 97.5% " << quantile(quiz, 0.025) << " to "
+ << quantile(complement(quiz, 0.025)) << endl; // 2.5 to 97.5%
+ cout << "2 to 98% " << quantile(quiz, 0.02) << " to "
+ << quantile(complement(quiz, 0.02)) << endl; // 2 to 98%
+
+ cout << "If guessing then percentiles 1 to 99% will get " << quantile(quiz, 0.01)
+ << " to " << quantile(complement(quiz, 0.01)) << " right." << endl;
+/*`
+Notice that these output integral values because the default policy is `integer_round_outwards`.
+[pre
+Quartiles 2 to 5
+1 standard deviation 2 to 5
+Deciles 1 to 6
+5 to 95% 0 to 7
+2.5 to 97.5% 0 to 8
+2 to 98% 0 to 8
+]
+*/
+
+//] [/binomial_quiz_example2]
+
+//[discrete_quantile_real
+/*`
+Quantiles values are controlled by the __understand_dis_quant quantile policy chosen.
+The default is `integer_round_outwards`,
+so the lower quantile is rounded down, and the upper quantile is rounded up.
+
+But we might believe that the real values tell us a little more - see __math_discrete.
+
+We could control the policy for *all* distributions by
+
+ #define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
+
+ at the head of the program would make this policy apply
+to this *one, and only*, translation unit.
+
+Or we can now create a (typedef for) policy that has discrete quantiles real
+(here avoiding any 'using namespaces ...' statements):
+*/
+ using boost::math::policies::policy;
+ using boost::math::policies::discrete_quantile;
+ using boost::math::policies::real;
+ using boost::math::policies::integer_round_outwards; // Default.
+ typedef boost::math::policies::policy<discrete_quantile<real> > real_quantile_policy;
+/*`
+Add a custom binomial distribution called ``real_quantile_binomial`` that uses ``real_quantile_policy``
+*/
+ using boost::math::binomial_distribution;
+ typedef binomial_distribution<double, real_quantile_policy> real_quantile_binomial;
+/*`
+Construct an object of this custom distribution:
+*/
+ real_quantile_binomial quiz_real(questions, success_fraction);
+/*`
+And use this to show some quantiles - that now have real rather than integer values.
+*/
+ cout << "Quartiles " << quantile(quiz, 0.25) << " to "
+ << quantile(complement(quiz_real, 0.25)) << endl; // Quartiles 2 to 4.6212
+ cout << "1 standard deviation " << quantile(quiz_real, 0.33) << " to "
+ << quantile(quiz_real, 0.67) << endl; // 1 sd 2.6654 4.194
+ cout << "Deciles " << quantile(quiz_real, 0.1) << " to "
+ << quantile(complement(quiz_real, 0.1))<< endl; // Deciles 1.3487 5.7583
+ cout << "5 to 95% " << quantile(quiz_real, 0.05) << " to "
+ << quantile(complement(quiz_real, 0.05))<< endl; // 5 to 95% 0.83739 6.4559
+ cout << "2.5 to 97.5% " << quantile(quiz_real, 0.025) << " to "
+ << quantile(complement(quiz_real, 0.025)) << endl; // 2.5 to 97.5% 0.42806 7.0688
+ cout << "2 to 98% " << quantile(quiz_real, 0.02) << " to "
+ << quantile(complement(quiz_real, 0.02)) << endl; // 2 to 98% 0.31311 7.7880
+
+ cout << "If guessing, then percentiles 1 to 99% will get " << quantile(quiz_real, 0.01)
+ << " to " << quantile(complement(quiz_real, 0.01)) << " right." << endl;
+/*`
+[pre
+Real Quantiles
+Quartiles 2 to 4.621
+1 standard deviation 2.665 to 4.194
+Deciles 1.349 to 5.758
+5 to 95% 0.8374 to 6.456
+2.5 to 97.5% 0.4281 to 7.069
+2 to 98% 0.3131 to 7.252
+If guessing then percentiles 1 to 99% will get 0 to 7.788 right.
+]
+*/
+
+//] [/discrete_quantile_real]
+ }
+ catch(const std::exception& e)
+ { // Always useful to include try & catch blocks because
+ // default policies are to throw exceptions on arguments that cause
+ // errors like underflow, overflow.
+ // Lacking try & catch blocks, the program will abort without a message below,
+ // which may give some helpful clues as to the cause of the exception.
+ std::cout <<
+ "\n""Message from thrown exception was:\n " << e.what() << std::endl;
+ }
+ return 0;
+} // int main()
+
+
+
+/*
+
+Output is:
+
+BAutorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\binomial_quiz_example.exe"
+Binomial distribution example - guessing in a quiz.
+In a quiz with 16 questions and with a probability of guessing right of 25 % or 1 in 4
+Probability of getting none right is 0.0100226
+Probability of getting exactly one right is 0.0534538
+Probability of getting exactly two right is 0.133635
+Probability of getting exactly 11 answers right by chance is 0.000247132
+Probability of getting all 16 answers right by chance is 2.32831e-010
+Guessed Probability
+ 0 0.0100226
+ 1 0.0534538
+ 2 0.133635
+ 3 0.207876
+ 4 0.225199
+ 5 0.180159
+ 6 0.110097
+ 7 0.0524273
+ 8 0.0196602
+ 9 0.00582526
+10 0.00135923
+11 0.000247132
+12 3.43239e-005
+13 3.5204e-006
+14 2.51457e-007
+15 1.11759e-008
+16 2.32831e-010
+Probability of getting none or one right is 0.0634764
+Probability of getting none or one right is 0.0634764
+Probability of getting <= 10 right (to fail) is 0.999715
+Probability of getting > 10 right (to pass) is 0.000285239
+Probability of getting > 10 right (to pass) is 0.000285239
+Probability of getting less than 11 (< 11) answers right by guessing is 0.999715
+Probability of getting at least 11(>= 11) answers right by guessing is 0.000285239, only 1 in 3505.83
+At most (<=)
+Guessed OK Probability
+ 0 0.01002259576
+ 1 0.0634764398
+ 2 0.1971110499
+ 3 0.4049871101
+ 4 0.6301861752
+ 5 0.8103454274
+ 6 0.9204427481
+ 7 0.9728700437
+ 8 0.9925302796
+ 9 0.9983555346
+10 0.9997147608
+11 0.9999618928
+12 0.9999962167
+13 0.9999997371
+14 0.9999999886
+15 0.9999999998
+16 1
+At least (>)
+Guessed OK Probability
+ 0 0.9899774042
+ 1 0.9365235602
+ 2 0.8028889501
+ 3 0.5950128899
+ 4 0.3698138248
+ 5 0.1896545726
+ 6 0.07955725188
+ 7 0.02712995629
+ 8 0.00746972044
+ 9 0.001644465374
+10 0.0002852391917
+11 3.810715862e-005
+12 3.783265129e-006
+13 2.628657967e-007
+14 1.140870154e-008
+15 2.328306437e-010
+16 0
+Probability of getting between 3 and 5 answers right by guessing is 0.6132
+Probability of getting between 3 and 5 answers right by guessing is 0.6132
+Probability of getting between 1 and 6 answers right by guessing is 0.9104
+Probability of getting between 1 and 8 answers right by guessing is 0.9825
+Probability of getting between 4 and 4 answers right by guessing is 0.2252
+By guessing, on average, one can expect to get 4 correct answers.
+Standard deviation is 1.732
+So about 2/3 will lie within 1 standard deviation and get between 3 and 5 correct.
+Mode (the most frequent) is 4
+Skewness is 0.2887
+Quartiles 2 to 5
+1 standard deviation 2 to 5
+Deciles 1 to 6
+5 to 95% 0 to 7
+2.5 to 97.5% 0 to 8
+2 to 98% 0 to 8
+If guessing then percentiles 1 to 99% will get 0 to 8 right.
+Quartiles 2 to 4.621
+1 standard deviation 2.665 to 4.194
+Deciles 1.349 to 5.758
+5 to 95% 0.8374 to 6.456
+2.5 to 97.5% 0.4281 to 7.069
+2 to 98% 0.3131 to 7.252
+If guessing, then percentiles 1 to 99% will get 0 to 7.788 right.
+
+*/
+
generated by cgit v1.2.3 (git 2.39.1) at 2024-12-30 14:44:28 +0000