binomial_quiz_example.cpp

Boost provides free peer-reviewed portable C++ source libraries. We emphasize libraries that work

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
[link math_toolkit.policy.pol_ref.discrete_quant_ref discrete 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
[link math_toolkit.policy.pol_tutorial.understand_dis_quant Understanding Discrete Quantile Policy].

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:

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 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.

*/