text: // test_poisson.cpp
  
  // Copyright Paul A. Bristow 2007.
  // 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)
  
  // Basic sanity test for Poisson Cumulative Distribution Function.
  
  #define BOOST_MATH_DISCRETE_QUANTILE_POLICY real
  
  #if !defined(TEST_FLOAT) && !defined(TEST_DOUBLE) && !defined(TEST_LDOUBLE) && !defined(TEST_REAL_CONCEPT)
  #  define TEST_FLOAT
  #  define TEST_DOUBLE
  #  define TEST_LDOUBLE
  #  define TEST_REAL_CONCEPT
  #endif
  
  #ifdef _MSC_VER
  #  pragma warning(disable: 4127) // conditional expression is constant.
  #endif
  
  #define BOOST_TEST_MAIN
  #include <boost/test/unit_test.hpp> // Boost.Test
  #include <boost/test/tools/floating_point_comparison.hpp>
  
  #include <boost/math/concepts/real_concept.hpp> // for real_concept
  #include <boost/math/distributions/poisson.hpp>
      using boost::math::poisson_distribution;
  #include <boost/math/tools/test.hpp> // for real_concept
  
  #include <boost/math/special_functions/gamma.hpp> // for (incomplete) gamma.
  //   using boost::math::qamma_Q;
  #include "table_type.hpp"
  #include "test_out_of_range.hpp"
  
  #include <iostream>
     using std::cout;
     using std::endl;
     using std::setprecision;
     using std::showpoint;
     using std::ios;
  #include <limits>
    using std::numeric_limits;
  
  template <class RealType> // Any floating-point type RealType.
  void test_spots(RealType)
  {
    // Basic sanity checks, tolerance is about numeric_limits<RealType>::digits10 decimal places,
     // guaranteed for type RealType, eg 6 for float, 15 for double,
     // expressed as a percentage (so -2) for BOOST_CHECK_CLOSE,
  
     int decdigits = numeric_limits<RealType>::digits10;
    // May eb >15 for 80 and 128-bit FP types.
    if (decdigits <= 0)
    { // decdigits is not defined, for example real concept,
      // so assume precision of most test data is double (for example, MathCAD).
       decdigits = numeric_limits<double>::digits10; // == 15 for 64-bit
    }
    if (decdigits > 15 ) // numeric_limits<double>::digits10)
    { // 15 is the accuracy of the MathCAD test data.
      decdigits = 15; // numeric_limits<double>::digits10;
    }
  
     decdigits -= 1; // Perhaps allow some decimal digit(s) margin of numerical error.
     RealType tolerance = static_cast<RealType>(std::pow(10., static_cast<double>(2-decdigits))); // 1e-6 (-2 so as %)
     tolerance *= 2; // Allow some bit(s) small margin (2 means + or - 1 bit) of numerical error.
     // Typically 2e-13% = 2e-15 as fraction for double.
  
     // Sources of spot test values:
  
    // Many be some combinations for which the result is 'exact',
    // or at least is good to 40 decimal digits.
     // 40 decimal digits includes 128-bit significand User Defined Floating-Point types,
     
     // Best source of accurate values is:
     // Mathworld online calculator (40 decimal digits precision, suitable for up to 128-bit significands)
     // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=GammaRegularized
     // GammaRegularized is same as gamma incomplete, gamma or gamma_q(a, x) or Q(a, z).
  
    // http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/PoissonDistribution.html
  
    // MathCAD defines ppois(k, lambda== mean) as k integer, k >=0.
    // ppois(0, 5) =  6.73794699908547e-3
    // ppois(1, 5) = 0.040427681994513;
    // ppois(10, 10) = 5.830397501929850E-001
    // ppois(10, 1) = 9.999999899522340E-001
    // ppois(5,5) = 0.615960654833065
  
    // qpois returns inverse Poisson distribution, that is the smallest (floor) k so that ppois(k, lambda) >= p
    // p is real number, real mean lambda > 0
    // k is approximately the integer for which probability(X <= k) = p
    // when random variable X has the Poisson distribution with parameters lambda.
    // Uses discrete bisection.
    // qpois(6.73794699908547e-3, 5) = 1
    // qpois(0.040427681994513, 5) = 
  
    // Test Poisson with spot values from MathCAD 'known good'.
  
    using boost::math::poisson_distribution;
    using  ::boost::math::poisson;
    using  ::boost::math::cdf;
    using  ::boost::math::pdf;
  
     // Check that bad arguments throw.
     BOOST_MATH_CHECK_THROW(
     cdf(poisson_distribution<RealType>(static_cast<RealType>(0)), // mean zero is bad.
        static_cast<RealType>(0)),  // even for a good k.
        std::domain_error); // Expected error to be thrown.
  
      BOOST_MATH_CHECK_THROW(
     cdf(poisson_distribution<RealType>(static_cast<RealType>(-1)), // mean negative is bad.
        static_cast<RealType>(0)),
        std::domain_error);
  
     BOOST_MATH_CHECK_THROW(
     cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unit OK,
        static_cast<RealType>(-1)),  // but negative events is bad.
        std::domain_error);
  
    BOOST_MATH_CHECK_THROW(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(0)), // mean zero is bad.
        static_cast<RealType>(99999)),  // for any k events. 
        std::domain_error);
    
    BOOST_MATH_CHECK_THROW(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(0)), // mean zero is bad.
        static_cast<RealType>(99999)),  // for any k events. 
        std::domain_error);
  
    BOOST_MATH_CHECK_THROW(
       quantile(poisson_distribution<RealType>(static_cast<RealType>(0)), // mean zero.
        static_cast<RealType>(0.5)),  // probability OK. 
        std::domain_error);
  
    BOOST_MATH_CHECK_THROW(
       quantile(poisson_distribution<RealType>(static_cast<RealType>(-1)), 
        static_cast<RealType>(-1)),  // bad probability. 
        std::domain_error);
  
    BOOST_MATH_CHECK_THROW(
       quantile(poisson_distribution<RealType>(static_cast<RealType>(1)), 
        static_cast<RealType>(-1)),  // bad probability. 
        std::domain_error);
  
    BOOST_MATH_CHECK_THROW(
       quantile(poisson_distribution<RealType>(static_cast<RealType>(1)), 
        static_cast<RealType>(1)),  // bad probability. 
        std::overflow_error);
  
    BOOST_MATH_CHECK_THROW(
       quantile(complement(poisson_distribution<RealType>(static_cast<RealType>(1)), 
        static_cast<RealType>(0))),  // bad probability. 
        std::overflow_error);
  
    BOOST_CHECK_EQUAL(
       quantile(poisson_distribution<RealType>(static_cast<RealType>(1)), 
        static_cast<RealType>(0)),  // bad probability. 
        0);
  
    BOOST_CHECK_EQUAL(
       quantile(complement(poisson_distribution<RealType>(static_cast<RealType>(1)), 
        static_cast<RealType>(1))),  // bad probability. 
        0);
  
    // Check some test values.
  
    BOOST_CHECK_CLOSE( // mode
       mode(poisson_distribution<RealType>(static_cast<RealType>(4))), // mode = mean = 4.
        static_cast<RealType>(4), // mode.
           tolerance);
  
    //BOOST_CHECK_CLOSE( // mode
    //   median(poisson_distribution<RealType>(static_cast<RealType>(4))), // mode = mean = 4.
    //    static_cast<RealType>(4), // mode.
        //   tolerance);
    poisson_distribution<RealType> dist4(static_cast<RealType>(40));
  
    BOOST_CHECK_CLOSE( // median
       median(dist4), // mode = mean = 4. median = 40.328333333333333 
        quantile(dist4, static_cast<RealType>(0.5)), // 39.332839138842637
           tolerance);
  
    // PDF
    BOOST_CHECK_CLOSE(
       pdf(poisson_distribution<RealType>(static_cast<RealType>(4)), // mean 4.
        static_cast<RealType>(0)),   
        static_cast<RealType>(1.831563888873410E-002), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       pdf(poisson_distribution<RealType>(static_cast<RealType>(4)), // mean 4.
        static_cast<RealType>(2)),   
        static_cast<RealType>(1.465251111098740E-001), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       pdf(poisson_distribution<RealType>(static_cast<RealType>(20)), // mean big.
        static_cast<RealType>(1)),   //  k small
        static_cast<RealType>(4.122307244877130E-008), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       pdf(poisson_distribution<RealType>(static_cast<RealType>(4)), // mean 4.
        static_cast<RealType>(20)),   //  K>> mean 
        static_cast<RealType>(8.277463646553730E-009), // probability.
           tolerance);
    
    // CDF
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(0)),  // zero k events. 
        static_cast<RealType>(3.678794411714420E-1), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(1)),  // one k event. 
        static_cast<RealType>(7.357588823428830E-1), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(2)),  // two k events. 
        static_cast<RealType>(9.196986029286060E-1), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(10)),  // two k events. 
        static_cast<RealType>(9.999999899522340E-1), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(15)),  // two k events. 
        static_cast<RealType>(9.999999999999810E-1), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(16)),  // two k events. 
        static_cast<RealType>(9.999999999999990E-1), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(17)),  // two k events. 
        static_cast<RealType>(1.), // probability unity for double.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(33)),  // k events at limit for float unchecked_factorial table. 
        static_cast<RealType>(1.), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(100)), // mean 100.
        static_cast<RealType>(33)),  // k events at limit for float unchecked_factorial table. 
        static_cast<RealType>(6.328271240363390E-15), // probability is tiny.
           tolerance * static_cast<RealType>(2e11)); // 6.3495253382825722e-015 MathCAD
        // Note that there two tiny probability are much more different.
  
     BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(100)), // mean 100.
        static_cast<RealType>(34)),  // k events at limit for float unchecked_factorial table. 
        static_cast<RealType>(1.898481372109020E-14), // probability is tiny.
           tolerance*static_cast<RealType>(2e11)); //         1.8984813721090199e-014 MathCAD
  
  
   BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(33)), // mean = k
        static_cast<RealType>(33)),  // k events above limit for float unchecked_factorial table. 
        static_cast<RealType>(5.461191812386560E-1), // probability.
           tolerance);
  
   BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(33)), // mean = k-1
        static_cast<RealType>(34)),  // k events above limit for float unchecked_factorial table. 
        static_cast<RealType>(6.133535681502950E-1), // probability.
           tolerance);
  
   BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1)), // mean unity.
        static_cast<RealType>(34)),  // k events above limit for float unchecked_factorial table. 
        static_cast<RealType>(1.), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(5.)), // mean
        static_cast<RealType>(5)),  // k events. 
        static_cast<RealType>(0.615960654833065), // probability.
           tolerance);
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(5.)), // mean
        static_cast<RealType>(1)),  // k events. 
        static_cast<RealType>(0.040427681994512805), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(5.)), // mean
        static_cast<RealType>(0)),  // k events (uses special case formula, not gamma). 
        static_cast<RealType>(0.006737946999085467), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(1.)), // mean
        static_cast<RealType>(0)),  // k events (uses special case formula, not gamma). 
        static_cast<RealType>(0.36787944117144233), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(10.)), // mean
        static_cast<RealType>(10)),  // k events. 
        static_cast<RealType>(0.5830397501929856), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(4.)), // mean
        static_cast<RealType>(5)),  // k events. 
        static_cast<RealType>(0.785130387030406), // probability.
           tolerance);
  
    // complement CDF
    BOOST_CHECK_CLOSE( // Complement CDF
       cdf(complement(poisson_distribution<RealType>(static_cast<RealType>(4.)), // mean
        static_cast<RealType>(5))),  // k events. 
        static_cast<RealType>(1 - 0.785130387030406), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE( // Complement CDF
       cdf(complement(poisson_distribution<RealType>(static_cast<RealType>(4.)), // mean
        static_cast<RealType>(0))),  // Zero k events (uses special case formula, not gamma).
        static_cast<RealType>(0.98168436111126578), // probability.
           tolerance);
    BOOST_CHECK_CLOSE( // Complement CDF
       cdf(complement(poisson_distribution<RealType>(static_cast<RealType>(1.)), // mean
        static_cast<RealType>(0))),  // Zero k events (uses special case formula, not gamma).
        static_cast<RealType>(0.63212055882855767), // probability.
           tolerance);
  
    // Example where k is bigger than max_factorial (>34 for float)
    // (therefore using log gamma so perhaps less accurate).
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(40.)), // mean
        static_cast<RealType>(40)),  // k events. 
        static_cast<RealType>(0.5419181783625430), // probability.
           tolerance);
  
     // Quantile & complement.
    BOOST_CHECK_CLOSE(
      boost::math::quantile(
           poisson_distribution<RealType>(5),  // mean.
           static_cast<RealType>(0.615960654833065)),  //  probability.
           static_cast<RealType>(5.), // Expect k = 5
           tolerance/5); // 
  
    // EQUAL is too optimistic - fails [5.0000000000000124 != 5]
    // BOOST_CHECK_EQUAL(boost::math::quantile( // 
    //       poisson_distribution<RealType>(5.),  // mean.
    //       static_cast<RealType>(0.615960654833065)),  //  probability.
    //       static_cast<RealType>(5.)); // Expect k = 5 events.
   
    BOOST_CHECK_CLOSE(boost::math::quantile(
           poisson_distribution<RealType>(4),  // mean.
           static_cast<RealType>(0.785130387030406)),  //  probability.
           static_cast<RealType>(5.), // Expect k = 5 events.
           tolerance/5); 
  
    // Check on quantile of other examples of inverse of cdf.
    BOOST_CHECK_CLOSE( 
       cdf(poisson_distribution<RealType>(static_cast<RealType>(10.)), // mean
        static_cast<RealType>(10)),  // k events. 
        static_cast<RealType>(0.5830397501929856), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(boost::math::quantile( // inverse of cdf above.
           poisson_distribution<RealType>(10.),  // mean.
           static_cast<RealType>(0.5830397501929856)),  //  probability.
           static_cast<RealType>(10.), // Expect k = 10 events.
           tolerance/5); 
  
  
    BOOST_CHECK_CLOSE(
       cdf(poisson_distribution<RealType>(static_cast<RealType>(4.)), // mean
        static_cast<RealType>(5)),  // k events. 
        static_cast<RealType>(0.785130387030406), // probability.
           tolerance);
  
    BOOST_CHECK_CLOSE(boost::math::quantile( // inverse of cdf above.
           poisson_distribution<RealType>(4.),  // mean.
           static_cast<RealType>(0.785130387030406)),  //  probability.
           static_cast<RealType>(5.), // Expect k = 10 events.
           tolerance/5); 
  
  
  
    //BOOST_CHECK_CLOSE(boost::math::quantile(
    //       poisson_distribution<RealType>(5),  // mean.
    //       static_cast<RealType>(0.785130387030406)),  //  probability.
    //        // 6.1882832344329559 result but MathCAD givest smallest integer ppois(k, mean) >= prob
    //       static_cast<RealType>(6.), // Expect k = 6 events. 
    //       tolerance/5); 
  
    //BOOST_CHECK_CLOSE(boost::math::quantile(
    //       poisson_distribution<RealType>(5),  // mean.
    //       static_cast<RealType>(0.77)),  //  probability.
    //        // 6.1882832344329559 result but MathCAD givest smallest integer ppois(k, mean) >= prob
    //       static_cast<RealType>(7.), // Expect k = 6 events. 
    //       tolerance/5); 
  
    //BOOST_CHECK_CLOSE(boost::math::quantile(
    //       poisson_distribution<RealType>(5),  // mean.
    //       static_cast<RealType>(0.75)),  //  probability.
    //        // 6.1882832344329559 result but MathCAD givest smallest integer ppois(k, mean) >= prob
    //       static_cast<RealType>(6.), // Expect k = 6 events. 
    //       tolerance/5); 
  
    BOOST_CHECK_CLOSE(
      boost::math::quantile(
           complement(
             poisson_distribution<RealType>(4),
             static_cast<RealType>(1 - 0.785130387030406))),  // complement.
             static_cast<RealType>(5), // Expect k = 5 events.
           tolerance/5);
  
    BOOST_CHECK_EQUAL(boost::math::quantile( // Check case when probability < cdf(0) (== pdf(0))
           poisson_distribution<RealType>(1),  // mean is small, so cdf and pdf(0) are about 0.35.
           static_cast<RealType>(0.0001)),  //  probability < cdf(0).
           static_cast<RealType>(0)); // Expect k = 0 events exactly.
            
    BOOST_CHECK_EQUAL(
      boost::math::quantile(
           complement(
             poisson_distribution<RealType>(1),
             static_cast<RealType>(0.9999))),  // complement, so 1-probability < cdf(0)
             static_cast<RealType>(0)); // Expect k = 0 events exactly.
  
    //
    // Test quantile policies against test data:
    //
  #define T RealType
  #include "poisson_quantile.ipp"
  
    for(unsigned i = 0; i < poisson_quantile_data.size(); ++i)
    {
       using namespace boost::math::policies;
       typedef policy<discrete_quantile<boost::math::policies::real> > P1;
       typedef policy<discrete_quantile<integer_round_down> > P2;
       typedef policy<discrete_quantile<integer_round_up> > P3;
       typedef policy<discrete_quantile<integer_round_outwards> > P4;
       typedef policy<discrete_quantile<integer_round_inwards> > P5;
       typedef policy<discrete_quantile<integer_round_nearest> > P6;
       RealType tol = boost::math::tools::epsilon<RealType>() * 20;
       if(!boost::is_floating_point<RealType>::value)
          tol *= 7;
       //
       // Check full real value first:
       //
       poisson_distribution<RealType, P1> p1(poisson_quantile_data[i][0]);
       RealType x = quantile(p1, poisson_quantile_data[i][1]);
       BOOST_CHECK_CLOSE_FRACTION(x, poisson_quantile_data[i][2], tol);
       x = quantile(complement(p1, poisson_quantile_data[i][1]));
       BOOST_CHECK_CLOSE_FRACTION(x, poisson_quantile_data[i][3], tol * 3);
       //
       // Now with round down to integer:
       //
       poisson_distribution<RealType, P2> p2(poisson_quantile_data[i][0]);
       x = quantile(p2, poisson_quantile_data[i][1]);
       BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][2]));
       x = quantile(complement(p2, poisson_quantile_data[i][1]));
       BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][3]));
       //
       // Now with round up to integer:
       //
       poisson_distribution<RealType, P3> p3(poisson_quantile_data[i][0]);
       x = quantile(p3, poisson_quantile_data[i][1]);
       BOOST_CHECK_EQUAL(x, ceil(poisson_quantile_data[i][2]));
       x = quantile(complement(p3, poisson_quantile_data[i][1]));
       BOOST_CHECK_EQUAL(x, ceil(poisson_quantile_data[i][3]));
       //
       // Now with round to integer "outside":
       //
       poisson_distribution<RealType, P4> p4(poisson_quantile_data[i][0]);
       x = quantile(p4, poisson_quantile_data[i][1]);
       BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? floor(poisson_quantile_data[i][2]) : ceil(poisson_quantile_data[i][2]));
       x = quantile(complement(p4, poisson_quantile_data[i][1]));
       BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? ceil(poisson_quantile_data[i][3]) : floor(poisson_quantile_data[i][3]));
       //
       // Now with round to integer "inside":
       //
       poisson_distribution<RealType, P5> p5(poisson_quantile_data[i][0]);
       x = quantile(p5, poisson_quantile_data[i][1]);
       BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? ceil(poisson_quantile_data[i][2]) : floor(poisson_quantile_data[i][2]));
       x = quantile(complement(p5, poisson_quantile_data[i][1]));
       BOOST_CHECK_EQUAL(x, poisson_quantile_data[i][1] < 0.5f ? floor(poisson_quantile_data[i][3]) : ceil(poisson_quantile_data[i][3]));
       //
       // Now with round to nearest integer:
       //
       poisson_distribution<RealType, P6> p6(poisson_quantile_data[i][0]);
       x = quantile(p6, poisson_quantile_data[i][1]);
       BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][2] + 0.5f));
       x = quantile(complement(p6, poisson_quantile_data[i][1]));
       BOOST_CHECK_EQUAL(x, floor(poisson_quantile_data[i][3] + 0.5f));
    }
     check_out_of_range<poisson_distribution<RealType> >(1);
  } // template <class RealType>void test_spots(RealType)
  
  //
  
  BOOST_AUTO_TEST_CASE( test_main )
  {
    // Check that can construct normal distribution using the two convenience methods:
    using namespace boost::math;
    poisson myp1(2); // Using typedef
     poisson_distribution<> myp2(2); // Using default RealType double.
  
     // Basic sanity-check spot values.
  
    // Some plain double examples & tests:
    cout.precision(17); // double max_digits10
    cout.setf(ios::showpoint);
    
    poisson mypoisson(4.); // // mean = 4, default FP type is double.
    cout << "mean(mypoisson, 4.) == " << mean(mypoisson) << endl;
    cout << "mean(mypoisson, 0.) == " << mean(mypoisson) << endl;
    cout << "cdf(mypoisson, 2.) == " << cdf(mypoisson, 2.) << endl;
    cout << "pdf(mypoisson, 2.) == " << pdf(mypoisson, 2.) << endl;
    
    // poisson mydudpoisson(0.);
    // throws (if BOOST_MATH_DOMAIN_ERROR_POLICY == throw_on_error).
  
  #ifndef BOOST_NO_EXCEPTIONS
    BOOST_MATH_CHECK_THROW(poisson mydudpoisson(-1), std::domain_error);// Mean must be > 0.
    BOOST_MATH_CHECK_THROW(poisson mydudpoisson(-1), std::logic_error);// Mean must be > 0.
  #else
    BOOST_MATH_CHECK_THROW(poisson(-1), std::domain_error);// Mean must be > 0.
    BOOST_MATH_CHECK_THROW(poisson(-1), std::logic_error);// Mean must be > 0.
  #endif
    // Passes the check because logic_error is a parent????
    // BOOST_MATH_CHECK_THROW(poisson mydudpoisson(-1), std::overflow_error); // fails the check
    // because overflow_error is unrelated - except from std::exception
    BOOST_MATH_CHECK_THROW(cdf(mypoisson, -1), std::domain_error); // k must be >= 0
  
    BOOST_CHECK_EQUAL(mean(mypoisson), 4.);
    BOOST_CHECK_CLOSE(
    pdf(mypoisson, 2.),  // k events = 2. 
      1.465251111098740E-001, // probability.
        5e-13);
  
    BOOST_CHECK_CLOSE(
    cdf(mypoisson, 2.),  // k events = 2. 
      0.238103305553545, // probability.
        5e-13);
  
  
  #if 0
    // Compare cdf from finite sum of pdf and gamma_q.
    using boost::math::cdf;
    using boost::math::pdf;
  
    double mean = 4.;
    cout.precision(17); // double max_digits10
    cout.setf(ios::showpoint);
    cout << showpoint << endl;  // Ensure trailing zeros are shown.
    // This also helps show the expected precision max_digits10
    //cout.unsetf(ios::showpoint); // No trailing zeros are shown.
  
    cout << "k          pdf                     sum                  cdf                   diff" << endl;
    double sum = 0.;
    for (int i = 0; i <= 50; i++)
    {
     cout << i << ' ' ;
     double p =  pdf(poisson_distribution<double>(mean), static_cast<double>(i));
     sum += p;
  
     cout << p << ' ' << sum << ' ' 
     << cdf(poisson_distribution<double>(mean), static_cast<double>(i)) << ' ';
       {
         cout << boost::math::gamma_q<double>(i+1, mean); // cdf
         double diff = boost::math::gamma_q<double>(i+1, mean) - sum; // cdf -sum
         cout << setprecision (2) << ' ' << diff; // 0 0 to 4, 1 eps 5 to 9, 10 to 20 2 eps, 21 upwards 3 eps
        
       }
      BOOST_CHECK_CLOSE(
      cdf(mypoisson, static_cast<double>(i)),
        sum, // of pdfs.
        4e-14); // Fails at 2e-14
     // This call puts the precision etc back to default 6 !!!
     cout << setprecision(17) << showpoint;
  
  
       cout << endl;
    }
  
     cout << cdf(poisson_distribution<double>(5), static_cast<double>(0)) << ' ' << endl; // 0.006737946999085467
     cout << cdf(poisson_distribution<double>(5), static_cast<double>(1)) << ' ' << endl; // 0.040427681994512805
     cout << cdf(poisson_distribution<double>(2), static_cast<double>(3)) << ' ' << endl; // 0.85712346049854715 
  
     { // Compare approximate formula in Wikipedia with quantile(half)
       for (int i = 1; i < 100; i++)
       {
         poisson_distribution<double> distn(static_cast<double>(i));
         cout << i << ' ' << median(distn) << ' ' << quantile(distn, 0.5) << ' ' 
           << median(distn) - quantile(distn, 0.5) << endl; // formula appears to be out-by-one??
       }  // so quantile(half) used via derived accressors.
     }
  #endif
  
     // (Parameter value, arbitrarily zero, only communicates the floating-point type).
  #ifdef TEST_POISSON
    test_spots(0.0F); // Test float.
  #endif
  #ifdef TEST_DOUBLE
    test_spots(0.0); // Test double.
  #endif
  #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
    if (numeric_limits<long double>::digits10 > numeric_limits<double>::digits10)
    { // long double is better than double (so not MSVC where they are same).
  #ifdef TEST_LDOUBLE
       test_spots(0.0L); // Test long double.
  #endif
    }
  
  #ifndef BOOST_MATH_NO_REAL_CONCEPT_TESTS
  #ifdef TEST_REAL_CONCEPT
    test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
  #endif
  #endif
  #endif
     
  } // BOOST_AUTO_TEST_CASE( test_main )
  
  /*
  
  Output:
  
  Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_poisson.exe"
  Running 1 test case...
  mean(mypoisson, 4.) == 4.0000000000000000
  mean(mypoisson, 0.) == 4.0000000000000000
  cdf(mypoisson, 2.) == 0.23810330555354431
  pdf(mypoisson, 2.) == 0.14652511110987343
  *** No errors detected
  
  */
  