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libs/math/example/neg_binomial_sample_sizes.cpp

```// neg_binomial_sample_sizes.cpp

// Copyright Paul A. Bristow 2007, 2010

// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.

#include <boost/math/distributions/negative_binomial.hpp>
using boost::math::negative_binomial;

// Default RealType is double so this permits use of:
double find_minimum_number_of_trials(
double k,     // number of failures (events), k >= 0.
double p,     // fraction of trails for which event occurs, 0 <= p <= 1.
double probability); // probability threshold, 0 <= probability <= 1.

#include <iostream>
using std::cout;
using std::endl;
using std::fixed;
using std::right;
#include <iomanip>
using std::setprecision;
using std::setw;

//[neg_binomial_sample_sizes

/*`
It centres around a routine that prints out a table of
minimum sample sizes (number of trials) for various probability thresholds:
*/
void find_number_of_trials(double failures, double p);
/*`
First define a table of significance levels: these are the maximum
acceptable probability that /failure/ or fewer events will be observed.
*/
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
/*`
Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence
that the desired number of failures will be observed.
The values range from a very low 0.5 or 50% confidence up to an extremely high
confidence of 99.999.

Much of the rest of the program is pretty-printing, the important part
is in the calculation of minimum number of trials required for each
value of alpha using:

(int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]);

find_minimum_number_of_trials returns a double,
so `ceil` rounds this up to ensure we have an integral minimum number of trials.
*/

void find_number_of_trials(double failures, double p)
{
// trials = number of trials
// failures = number of failures before achieving required success(es).
// p        = success fraction (0 <= p <= 1.).
//
// Calculate how many trials we need to ensure the
// required number of failures DOES exceed "failures".

cout << "\n""Target number of failures = " << (int)failures;
cout << ",   Success fraction = " << fixed << setprecision(1) << 100 * p << "%" << endl;
cout << "____________________________\n"
"Confidence        Min Number\n"
" Value (%)        Of Trials \n"
"____________________________\n";
// Now print out the data for the alpha table values.
for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
{ // Confidence values %:
cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]) << "      "
// find_minimum_number_of_trials
<< setw(6) << right
<< (int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]))
<< endl;
}
cout << endl;
} // void find_number_of_trials(double failures, double p)

/*` finally we can produce some tables of minimum trials for the chosen confidence levels:
*/

int main()
{
find_number_of_trials(5, 0.5);
find_number_of_trials(50, 0.5);
find_number_of_trials(500, 0.5);
find_number_of_trials(50, 0.1);
find_number_of_trials(500, 0.1);
find_number_of_trials(5, 0.9);

return 0;
} // int main()

//]  [/neg_binomial_sample_sizes.cpp end of Quickbook in C++ markup]

/*

Output is:
Target number of failures = 5,   Success fraction = 50.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000          11
75.000          14
90.000          17
95.000          18
99.000          22
99.900          27
99.990          31
99.999          36

Target number of failures = 50,   Success fraction = 50.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000         101
75.000         109
90.000         115
95.000         119
99.000         128
99.900         137
99.990         146
99.999         154

Target number of failures = 500,   Success fraction = 50.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000        1001
75.000        1023
90.000        1043
95.000        1055
99.000        1078
99.900        1104
99.990        1126
99.999        1146

Target number of failures = 50,   Success fraction = 10.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000          56
75.000          58
90.000          60
95.000          61
99.000          63
99.900          66
99.990          68
99.999          71

Target number of failures = 500,   Success fraction = 10.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000         556
75.000         562
90.000         567
95.000         570
99.000         576
99.900         583
99.990         588
99.999         594

Target number of failures = 5,   Success fraction = 90.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000          57
75.000          73
90.000          91
95.000         103
99.000         127
99.900         159
99.990         189
99.999         217

Target number of failures = 5,   Success fraction = 95.0%
____________________________
Confidence        Min Number
Value (%)        Of Trials
____________________________
50.000         114
75.000         148
90.000         184
95.000         208
99.000         259
99.900         324
99.990         384
99.999         442

*/
```