# Boost-Commit :

From: pbristow_at_[hidden]
Date: 2007-08-21 09:04:17

Author: pbristow
Date: 2007-08-21 09:04:16 EDT (Tue, 21 Aug 2007)
New Revision: 38816
URL: http://svn.boost.org/trac/boost/changeset/38816

Log:
First cut

sandbox/math_toolkit/libs/math/example/binomial_quiz_example.cpp (contents, props changed)

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+// Copyright Paul A. Bristow 2007
+
+// Use, modification and distribution are subject to the
+// Boost Software License, Version 1.0.
+
+// 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 quesses the answer to each question,
+ so the probability of getting a correct answer on any given question is 1/4 = 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.
+
+ The number of correct answers, X, is distributed as a binomial random variable
+ with binomial distribution parameters n = 16 and p = 0.25.
+ The program below displays the probabilities for each of the 17 possible outcomes,
+ i.e., for X = 0, 1, ..., 16, in a line chart.
+*/
+// #define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
+/*`
+For information on the effect of this define, see section on quantiles below.
+
+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;
+// ][/binomial_quiz_example1]
+
+//[binomial_confidence_limits
+void confidence_limits_on_frequency(unsigned trials, unsigned successes)
+{ // trials = Total number of trials, successes = Total number of observed successes.
+
+ // Calculate confidence limits for an observed
+ // frequency of occurrence that follows a binomial distribution.
+ // Print out general info:
+ cout <<
+ "___________________________________________\n"
+ "2-Sided Confidence Limits For Success Ratio\n"
+ "___________________________________________\n\n";
+ cout << setprecision(7);
+ cout << setw(40) << left << "Number of Observations"
+ << "= " << trials << "\n";
+ cout << setw(40) << left << "Number of successes"
+ << "= " << successes << "\n";
+ cout << setw(40) << left << "Sample frequency of occurrence"
+ << "= " << double(successes) / trials << "\n";
+
+ // Define a table of significance levels:
+ double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
+ //
+ //
+ cout << "\n\n"
+ "___________________________________________\n"
+ "Confidence Lower Upper\n"
+ " Value (%) Limit Limit\n"
+ "___________________________________________\n";
+
+ for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+ { // Now print out the data for the table rows.
+ // Confidence value:
+ cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
+ // Calculate bounds:
+ double l = binomial::estimate_lower_bound_on_p(trials, successes, alpha[i]/2);
+ double u = binomial::estimate_upper_bound_on_p(trials, successes, alpha[i]/2);
+ // And print limits:
+ cout << fixed << setprecision(5) << setw(15) << right << l;
+ cout << fixed << setprecision(5) << setw(15) << right << u << endl;
+ }
+ cout << endl;
+} // void confidence_limits_on_frequency()
+//] [/binomial_confidence_limits]
+
+//[binomial_quiz_example2
+int main()
+{
+ try
+ {
+ cout << "Binomial distribution example - guessing in a quiz." << endl;
+/*`The number of correct answers, X, is distributed as a binomial random variable
+with binomial distribution parameters n = 16 and p = 0.25.
+*/
+ cout.precision(5); // Might be able to calculate a best value for this?
+ int questions = 16;
+ double success_fraction = (double)answers / (double)questions; // If a random guess.
+ // Caution: = answers / questions would be zero (because they are integers)!
+ int pass_score = 11;
+
+/*`Construct our Binomial distribution.
+*/
+ binomial quiz(questions, success_fraction);
+/*`and display the parameters we used.
+*/
+ 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 some probabilities of just guessing: these don't give any encouragement to guessers!
+*/
+ cout << "Probability of getting none right is " << pdf(quiz, 0) << endl; // 0.010023
+ cout << "Probability of getting at least one right is " << 1 - pdf(quiz, 1) << endl; // 0.94655
+ cout << "Probability of getting none or one right is " << pdf(quiz, 0) + pdf(quiz, 1) << endl; // 0.063476
+ cout << "Probability of getting exactly one right is " << pdf(quiz, 1) << endl;
+ cout << "Probability of getting exactly 11 right is " << pdf(quiz, 11) << endl;
+ cout << "Probability of getting > 10 right (to pass) is " << cdf(complement(quiz, 10)) << endl;
+
+ // Using Binomial probabilities.
+ cout << "The probability of getting all the answers wrong by chance is "
+ << pdf(quiz, 0) << endl;
+ cout << "The probability of getting all the answers right by chance is "
+ << pdf(quiz, questions) << endl;
+ cout << "The probability of getting exactly " << pass_score
+ << " answers right by guessing is " << pdf(quiz, pass_score) << endl << endl;
+
+ cout << "The probability of getting less then " << pass_score
+ << "(< " << pass_score << ") answers right by guessing is "
+ << cdf(quiz, pass_score) << endl;
+
+ cout << "The 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;
+
+/*`Tabulate probability versus number right.
+*/
+ cout << "\n" "Guessed right Probability" << right << endl;
+ for (int successes = 0; successes <= questions; successes++)
+ {
+ double probability = pdf(quiz, successes);
+ cout << setw(2) << successes << " " << probability << endl;
+ }
+ cout << endl;
+
+
+ cout << "\n" "At most (<=)""\n""Guessed right Probability" << right << endl;
+ for (int score = 0; score <= questions; score++)
+ {
+ cout << setw(2) << score << " " << cdf(quiz, score) << endl;
+ }
+ cout << endl;
+
+ cout << "\n" "At least (>=)""\n""Guessed right Probability" << right << endl;
+ for (int score = 0; score <= questions; score++)
+ {
+ cout << setw(2) << score << " " << cdf(complement(quiz, score)) << endl;
+ }
+/*`Calculate the probability of getting a range of guesses right,
+first by adding the exact probabilities of each of low ... high.
+*/
+ int low = 3;
+ int high = 5;
+ double sum = 0.;
+ for (int i = low; i <= high; i++)
+ {
+ sum += pdf(quiz, i);
+ }
+ cout << "The probability of getting between "
+ << low << " and " << high << " answers right by guessing is "
+ << sum << endl; // 0.61323
+
+/*`Or, better, we can use the difference of cdfs instead:
+*/
+ cout << "The probability of getting between " << low << " and " << high << " answers right by guessing is "
+ << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // 0.61323
+ // And a few more combinations of high and low choices:
+ low = 1; high = 6;
+ cout << "The 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 << "The 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 << "The 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
+ low = 3; high = 5;
+ cout << "The probability of getting between " << low << " and " << high << " answers right by guessing is "
+ << cdf(quiz, high) - cdf(quiz, low - 1) << endl; // P 3 to 5 right
+
+/*`Using Binomial distribution moments,
+*/
+ 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;
+
+
+
+/*`Show the use of quantiles (percentiles or percentage points) for a few probability levels:
+*/
+ cout << "Quantiles" << endl;
+ cout << "Quartiles " << quantile(quiz, 0.25) << " to "
+ << quantile(complement(quiz, 0.25)) << endl; // Quartiles 2.2821 4.6212
+ cout << "1 sd " << quantile(quiz, 0.33) << " to "
+ << quantile(quiz, 0.67) << endl; // 1 sd 2.6654 4.1935
+ cout << "Deciles " << quantile(quiz, 0.1) << " to "
+ << quantile(complement(quiz, 0.1))<< endl; // Deciles 1.3487 5.7583
+ cout << "5 to 95% " << quantile(quiz, 0.05) << " to "
+ << quantile(complement(quiz, 0.05))<< endl; // 5 to 95% 0.83739 6.4559
+ cout << "2.5 to 97.5% " << quantile(quiz, 0.025) << " to "
+ << quantile(complement(quiz, 0.025)) << endl; // 2.5 to 97.5% 0.42806 7.0688
+ cout << "2 to 98% " << quantile(quiz, 0.02) << " to "
+ << quantile(complement(quiz, 0.02)) << endl; // 2 to 98% 0.31311 7.7880
+
+ cout << "If guessing then percentiles 1 to 99% will get " << quantile(quiz, 0.01)
+ << " to " << quantile(complement(quiz, 0.01)) << " right." << endl;
+
+//] [/binomial_quiz_example2]
+
+//[discrete_quantile_real
+/*`The quantiles values are controlled by the discrete quantile policy chosen.
+The default is integer_outside,
+so the lower quantile is rounded down, and the upper quantile is rounded up.
+
+We can 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 create a (typedef for) policy that has discrete quantiles real.
+*/using namespace boost::math::policies;
+/*`
+Convenient for all policy and typelist values like discrete_quantile.
+*/
+using namespace boost::math;
+/*`
+for binomial_distribution
+
+Or to be more specific, to avoid 'using namespaces ...' statements:
+*/
+ using boost::math::policies::policy;
+ using boost::math::policies::discrete_quantile;
+ using boost::math::policies::real;
+ using boost::math::policies::integer_outside; // Default.
+ typedef boost::math::policies::policy<discrete_quantile<real> > real_quantile_policy;
+/*`Add a 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 a distribution of this custom real_quantile_binomial 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 << "Real Quartiles " << quantile(quiz_real, 0.25) << " to "
+ << quantile(complement(quiz_real, 0.25)) << endl; // Real Quartiles 2.2821 to 4.6212
+
+//] [/discrete_quantile_real]
+
+//[binomial_quiz_example3
+
+/*`Now we can repeat the
+
+/*`Confidence intervals for various confidences 1 - alpha,
+probability as % = 100 * (1 - alpha[i]), so alpha 0.05 = 95% confidence.
+*/
+ double alpha[] = {0.5, 0.33, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001};
+
+ cout << "\n\n"
+ "Confidence % Lower Upper ""\n";
+ for (unsigned int i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
+ {
+ cout << fixed << setprecision(3) << setw(10) << right << 100 * (1 - alpha[i]);
+ double l = real_quantile_binomial::estimate_lower_bound_on_p(questions, answers, alpha[i]/2);
+ double u = real_quantile_binomial::estimate_upper_bound_on_p(questions, answers, alpha[i]/2);
+ cout << fixed << setprecision(5) << setw(15) << right << l;
+ cout << fixed << setprecision(5) << setw(15) << right << u << endl;
+ }
+ cout << endl;
+
+ // static RealType estimate_lower_bound_on_p(RealType trials, RealType successes, RealType probability)
+ int successes = 11;
+ cout << "Success fraction " << quiz.success_fraction() << ", "
+ << quiz.trials() << " correct needed, " << successes << " successes " << endl;
+ cout << "Lower bound = " << binomial::estimate_lower_bound_on_p(quiz.trials(), successes, 0.05) << endl; // 0.51560
+ // Bounds now 0.45165 to 0.86789 ????
+ // Or equivalently:
+ cout << "Lower bound = " << quiz.estimate_lower_bound_on_p(quiz.trials(), successes, 0.05) << endl; // 0.51560
+ cout << "Upper bound = " << quiz.estimate_upper_bound_on_p(quiz.trials(), successes, 0.05) << endl; // 0.86789
+
+ // Print a table of showing upper and lower bounds for a range of confidence levels.
+ confidence_limits_on_frequency(20, 2); // 20 trials, 2 successes
+ confidence_limits_on_frequency(200, 20);
+ confidence_limits_on_frequency(2000, 200);
+ }
+ 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()
+//] [/binomial_quiz_example3]
+
+
+/*
+
+Output is:
+
+Binomial distribution example - guessing in a quiz.
+In a quiz with 16 and with a probability of guessing right of 25 % or 1 in 4
+Probability of getting none right is 0.010023
+Probability of getting at least one right is 0.98998
+Probability of getting exactly one right is 0.053454
+Guessed right Probability
+ 0 0.010023
+ 1 0.053454
+ 2 0.13363
+ 3 0.20788
+ 4 0.2252
+ 5 0.18016
+ 6 0.1101
+ 7 0.052427
+ 8 0.01966
+ 9 0.0058253
+10 0.0013592
+11 0.00024713
+12 3.4324e-005
+13 3.5204e-006
+14 2.5146e-007
+15 1.1176e-008
+16 2.3283e-010
+By guessing, on average, one can expect to get 4 correct answers.
+Standard deviation is 1.7321
+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 0.28868
+The probability of getting all the answers wrong by chance is 0.010023
+The probability of getting all the answers right by chance is 2.3283e-010
+The probability of getting exactly 11 answers right by guessing is 0.00024713
+The probability of getting at most 11(<= 11) answers right by guessing is 0.99996
+The probability of getting at least 11(>= 11) answers right by guessing is 3.8107e-005
+At most (<=)
+Guessed right Probability
+ 0 0.010023
+ 1 0.063476
+ 2 0.19711
+ 3 0.40499
+ 4 0.63019
+ 5 0.81035
+ 6 0.92044
+ 7 0.97287
+ 8 0.99253
+ 9 0.99836
+10 0.99971
+11 0.99996
+12 1
+13 1
+14 1
+15 1
+16 1
+At least (>=)
+Guessed right Probability
+ 0 0.98998
+ 1 0.93652
+ 2 0.80289
+ 3 0.59501
+ 4 0.36981
+ 5 0.18965
+ 6 0.079557
+ 7 0.02713
+ 8 0.0074697
+ 9 0.0016445
+10 0.00028524
+11 3.8107e-005
+12 3.7833e-006
+13 2.6287e-007
+14 1.1409e-008
+15 2.3283e-010
+16 0
+The probability of getting between 3 and 5 answers right by guessing is 0.61323
+The probability of getting between 3 and 5 answers right by guessing is 0.61323
+The probability of getting between 1 and 6 answers right by guessing is 0.91042
+The probability of getting between 1 and 8 answers right by guessing is 0.98251
+The probability of getting between 4 and 4 answers right by guessing is 0.2252
+Quantiles
+Quartiles 2 to 5
+1 sd 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.
+Real Quartiles 2.2821 to 4.6212
+Using quantiles:
+ p plim
+0.01 0.010022595757618546
+ q qlim
+0.98999999999999999 -0.98997740424238145
+com 99% 0 0
+Confidence % Lower Upper
+ 50.000 0.16108 0.36424
+ 67.000 0.13651 0.40078
+ 75.000 0.12318 0.42258
+ 90.000 0.09025 0.48440
+ 95.000 0.07266 0.52377
+ 99.000 0.04545 0.59913
+ 99.900 0.02427 0.68125
+ 99.990 0.01329 0.74363
+ 99.999 0.00737 0.79236
+Success fraction 0.25000, 16.00000 trials
+Lower bound = 0.45165
+Lower bound = 0.45165
+Upper bound = 0.86789
+___________________________________________
+2-Sided Confidence Limits For Success Ratio
+___________________________________________
+Number of Observations = 20
+Number of successes = 2
+Sample frequency of occurrence = 0.1000000
+___________________________________________
+Confidence Lower Upper
+ Value (%) Limit Limit
+___________________________________________
+ 50.000 0.04812 0.18675
+ 75.000 0.03078 0.23163
+ 90.000 0.01807 0.28262
+ 95.000 0.01235 0.31698
+ 99.000 0.00530 0.38713
+ 99.900 0.00164 0.47093
+ 99.990 0.00051 0.54084
+ 99.999 0.00016 0.60020
+___________________________________________
+2-Sided Confidence Limits For Success Ratio
+___________________________________________
+Number of Observations = 200
+Number of successes = 20
+Sample frequency of occurrence = 0.1000000
+___________________________________________
+Confidence Lower Upper
+ Value (%) Limit Limit
+___________________________________________
+ 50.000 0.08462 0.11824
+ 75.000 0.07580 0.12959
+ 90.000 0.06726 0.14199
+ 95.000 0.06216 0.15021
+ 99.000 0.05293 0.16698
+ 99.900 0.04343 0.18756
+ 99.990 0.03641 0.20571
+ 99.999 0.03095 0.22226
+___________________________________________
+2-Sided Confidence Limits For Success Ratio
+___________________________________________
+Number of Observations = 2000
+Number of successes = 200
+Sample frequency of occurrence = 0.1000000
+___________________________________________
+Confidence Lower Upper
+ Value (%) Limit Limit
+___________________________________________
+ 50.000 0.09536 0.10491
+ 75.000 0.09228 0.10822
+ 90.000 0.08916 0.11172
+ 95.000 0.08720 0.11399
+ 99.000 0.08344 0.11850
+ 99.900 0.07921 0.12385
+ 99.990 0.07577 0.12845
+ 99.999 0.07282 0.13256
+Build Time 0:03
+
+*/
+