normal_misc_examples.cpp

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// negative_binomial_example3.cpp

// Copyright Paul A. Bristow 2007.

// 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)

// Example of using normal distribution.

// Note that this file contains Quickbook mark-up as well as code
// and comments, don't change any of the special comment mark-ups!

//[normal_basic1
/*`
First we need some includes to access the normal distribution
(and some std output of course).
*/

#include <boost/math/distributions/normal.hpp> // for normal_distribution
  using boost::math::normal; // typedef provides default type is double.

#include <iostream>
  using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;
#include <iomanip>
  using std::setw; using std::setprecision;
#include <limits>
  using std::numeric_limits;

int main()
{
  cout << "Example: Normal distribution, Miscellaneous Applications.";

  try
  {
    { // Traditional tables and values.
/*`Let's start by printing some traditional tables.
*/      
      double step = 1.; // in z 
      double range = 4; // min and max z = -range to +range.
      int precision = 17; // traditional tables are only computed to much lower precision.

      // Construct a standard normal distribution s
        normal s; // (default mean = zero, and standard deviation = unity)
        cout << "Standard normal distribution, mean = "<< s.mean()
          << ", standard deviation = " << s.standard_deviation() << endl;

/*` First the probability distribution function (pdf).
*/
      cout << "Probability distribution function values" << endl;
      cout << "  z " "      pdf " << endl;
      cout.precision(5);
      for (double z = -range; z < range + step; z += step)
      {
        cout << left << setprecision(3) << setw(6) << z << " " 
          << setprecision(precision) << setw(12) << pdf(s, z) << endl;
      }
      cout.precision(6); // default
      /*`And the area under the normal curve from -[infin] up to z,
      the cumulative distribution function (cdf).
*/
      // For a standard normal distribution 
      cout << "Standard normal mean = "<< s.mean()
        << ", standard deviation = " << s.standard_deviation() << endl;
      cout << "Integral (area under the curve) from - infinity up to z " << endl;
      cout << "  z " "      cdf " << endl;
      for (double z = -range; z < range + step; z += step)
      {
        cout << left << setprecision(3) << setw(6) << z << " " 
          << setprecision(precision) << setw(12) << cdf(s, z) << endl;
      }
      cout.precision(6); // default

/*`And all this you can do with a nanoscopic amount of work compared to
the team of *human computers* toiling with Milton Abramovitz and Irene Stegen
at the US National Bureau of Standards (now [@http://www.nist.gov NIST]). 
Starting in 1938, their "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables",
was eventually published in 1964, and has been reprinted numerous times since.
(A major replacement is planned at [@http://dlmf.nist.gov Digital Library of Mathematical Functions]).

Pretty-printing a traditional 2-dimensional table is left as an exercise for the student,
but why bother now that the Math Toolkit lets you write
*/
    double z = 2.; 
    cout << "Area for z = " << z << " is " << cdf(s, z) << endl; // to get the area for z.
/*`
Correspondingly, we can obtain the traditional 'critical' values for significance levels.
For the 95% confidence level, the significance level usually called alpha,
is 0.05 = 1 - 0.95 (for a one-sided test), so we can write
*/
     cout << "95% of area has a z below " << quantile(s, 0.95) << endl;
   // 95% of area has a z below 1.64485
/*`and a two-sided test (a comparison between two levels, rather than a one-sided test)

*/
     cout << "95% of area has a z between " << quantile(s, 0.975)
       << " and " << -quantile(s, 0.975) << endl;
   // 95% of area has a z between 1.95996 and -1.95996
/*`

First, define a table of significance levels: these are the probabilities
that the true occurrence frequency lies outside the calculated interval.

It is convenient to have an alpha level for the probability that z lies outside just one standard deviation.
This will not be some nice neat number like 0.05, but we can easily calculate it,
*/
    double alpha1 = cdf(s, -1) * 2; // 0.3173105078629142
    cout << setprecision(17) << "Significance level for z == 1 is " << alpha1 << endl;
/*`    
    and place in our array of favorite alpha values.
*/
    double alpha[] = {0.3173105078629142, // z for 1 standard deviation.
      0.20, 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 true occurrence frequency lies *inside* the calculated interval.

*/
    cout << "level of significance (alpha)" << setprecision(4) << endl;
    cout << "2-sided       1 -sided          z(alpha) " << endl;
    for (int i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
    {
      cout << setw(15) << alpha[i] << setw(15) << alpha[i] /2 << setw(10) << quantile(complement(s,  alpha[i]/2)) << endl;
      // Use quantile(complement(s, alpha[i]/2)) to avoid potential loss of accuracy from quantile(s,  1 - alpha[i]/2) 
    }
    cout << endl;

/*`Notice the distinction between one-sided (also called one-tailed)
where we are using a > *or* < test (and not both)
and considering the area of the tail (integral) from z up to +[infin],
and a two-sided test where we are using two > *and* < tests, and thus considering two tails,
from -[infin] up to z low and z high up to +[infin].

So the 2-sided values alpha[i] are calculated using alpha[i]/2.

If we consider a simple example of alpha = 0.05, then for a two-sided test,
the lower tail area from -[infin] up to -1.96 is 0.025 (alpha/2)
and the upper tail area from +z up to +1.96 is also  0.025 (alpha/2),
and the area between -1.96 up to 12.96 is alpha = 0.95.
and the sum of the two tails is 0.025 + 0.025 = 0.05,

*/
//] [/[normal_basic1]

//[normal_basic2

/*`Armed with the cumulative distribution function, we can easily calculate the
easy to remember proportion of values that lie within 1, 2 and 3 standard deviations from the mean.

*/
    cout.precision(3);
    cout << showpoint << "cdf(s, s.standard_deviation()) = "
      << cdf(s, s.standard_deviation()) << endl;  // from -infinity to 1 sd
    cout << "cdf(complement(s, s.standard_deviation())) = "
      << cdf(complement(s, s.standard_deviation())) << endl;
    cout << "Fraction 1 standard deviation within either side of mean is "
      << 1 -  cdf(complement(s, s.standard_deviation())) * 2 << endl;
    cout << "Fraction 2 standard deviations within either side of mean is "
      << 1 -  cdf(complement(s, 2 * s.standard_deviation())) * 2 << endl;
    cout << "Fraction 3 standard deviations within either side of mean is "
      << 1 -  cdf(complement(s, 3 * s.standard_deviation())) * 2 << endl;

/*`
To a useful precision, the 1, 2 & 3 percentages are 68, 95 and 99.7,
and these are worth memorising as useful 'rules of thumb', as, for example, in
[@http://en.wikipedia.org/wiki/Standard_deviation standard deviation]:

[pre
Fraction 1 standard deviation within either side of mean is 0.683
Fraction 2 standard deviations within either side of mean is 0.954
Fraction 3 standard deviations within either side of mean is 0.997
]

We could of course get some really accurate values for these
[@http://en.wikipedia.org/wiki/Confidence_interval confidence intervals]
by using cout.precision(15);

[pre 
Fraction 1 standard deviation within either side of mean is 0.682689492137086
Fraction 2 standard deviations within either side of mean is 0.954499736103642
Fraction 3 standard deviations within either side of mean is 0.997300203936740
]

But before you get too excited about this impressive precision,
don't forget that the *confidence intervals of the standard deviation* are surprisingly wide,
especially if you have estimated the standard deviation from only a few measurements.
*/
//] [/[normal_basic2]


//[normal_bulbs_example1
/*`
Examples from K. Krishnamoorthy, Handbook of Statistical Distributions with Applications,
ISBN 1 58488 635 8, page 125... implemented using the Math Toolkit library.

A few very simple examples are shown here:
*/
// K. Krishnamoorthy, Handbook of Statistical Distributions with Applications,
 // ISBN 1 58488 635 8, page 125, example 10.3.5
/*`Mean lifespan of 100 W bulbs is 1100 h with standard deviation of 100 h.
Assuming, perhaps with little evidence and much faith, that the distribution is normal,
we construct a normal distribution called /bulbs/ with these values:
*/
    double mean_life = 1100.;
    double life_standard_deviation = 100.;
    normal bulbs(mean_life, life_standard_deviation); 
    double expected_life = 1000.;

/*`The we can use the Cumulative distribution function to predict fractions
(or percentages, if * 100) that will last various lifetimes.
*/
    cout << "Fraction of bulbs that will last at best (<=) " // P(X <= 1000)
      << expected_life << " is "<< cdf(bulbs, expected_life) << endl;
    cout << "Fraction of bulbs that will last at least (>) " // P(X > 1000)
      << expected_life << " is "<< cdf(complement(bulbs, expected_life)) << endl;
    double min_life = 900;
    double max_life = 1200;
    cout << "Fraction of bulbs that will last between "
      << min_life << " and " << max_life << " is "
      << cdf(bulbs, max_life)  // P(X <= 1200)
       - cdf(bulbs, min_life) << endl; // P(X <= 900) 
/*`
[note Real-life failures are often very ab-normal,
with a significant number that 'dead-on-arrival' or suffer failure very early in their life:
the lifetime of the survivors of 'early mortality' may be well described by the normal distribution.]
*/
//] [/normal_bulbs_example1 Quickbook end]
  }
  { 
    // K. Krishnamoorthy, Handbook of Statistical Distributions with Applications,
    // ISBN 1 58488 635 8, page 125, Example 10.3.6  

//[normal_bulbs_example3
/*`Weekly demand for 5 lb sacks of onions at a store is normally distributed with mean 140 sacks and standard deviation 10.
*/
  double mean = 140.; // sacks per week.
  double standard_deviation = 10; 
  normal sacks(mean, standard_deviation);

  double stock = 160.; // per week.
  cout << "Percentage of weeks overstocked "
    << cdf(sacks, stock) * 100. << endl; // P(X <=160)
  // Percentage of weeks overstocked 97.7

/*`So there will be lots of mouldy onions!
So we should be able to say what stock level will meet demand 95% of the weeks.
*/
  double stock_95 = quantile(sacks, 0.95);
  cout << "Store should stock " << int(stock_95) << " sacks to meet 95% of demands." << endl;
/*`And it is easy to estimate how to meet 80% of demand, and waste even less.
*/
  double stock_80 = quantile(sacks, 0.80);
  cout << "Store should stock " << int(stock_80) << " sacks to meet 8 out of 10 demands." << endl;