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Evaluates the Bradford distribution CDF.

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Interface

#include <codecogs/statistics/distributions/continuous/bradford/cdf.h>

using namespace Statistics::Distributions::Continuous::Bradford;

double CDF(double x, double a, double b, double c)

This function evaluates the CDF of the Bradford distribution with given argument, defined by

$D(x, a, b, c) = \frac{ \ln \left( 1 + \frac{c(x - a)}{b - a} \right)} {\ln (c + 1)} \quad \mbox{for } a < x \leq b$

(1)

and

$D(x, a, b, c) = 0 \quad \mbox{for } x \leq a \qquad D(x, a, b, c) = 1 \quad \mbox{for } x > b$

(2)

where

$a < b \quad \mbox{ and } \quad c > 0$

(3)

In the following figure a =-10 b= 10 and the curve of the line is set by c =5 \graph x=-10:10 a=-10 b=10 c=5

Example 1

In the example that follows, the CDF is evaluated using values from 0 up to 0.8 with a step equal to 0.1. The maximum number of precision digits, implicitly set to 17, may be changed through the <em> PRECISION </em> define.

#include <codecogs/stats/dists/continuous/bradford/cdf.h>
#include <iostream>
#include <iomanip>

#define PRECISION 17

int main()
{
  std::cout << "The values of the Bradford CDF with " << std::endl;
  std::cout << "a = 0, b = 0.7, c = 3 and" << std::endl;
  std::cout << "x = {0, 0.1, 0.2, ... , 0.8} are" << std::endl;
  std::cout << std::endl;
  std::cout << std::setprecision(10);
  for (double x = 0; x < 0.81; x += 0.1)
  {
    std::cout << std::setprecision(1);
    std::cout << "x = " << std::setw(3) << x << " : ";
    std::cout << std::setprecision(PRECISION);
    std::cout << Stats::Dists::Continuous::Bradford::CDF(x, 0, 0.7, 3);
    std::cout << std::endl;
  }
  return 0;
}

Output

The values of the Bradford CDF with
a = 0, b = 0.7, c = 3 and
x = {0, 0.1, 0.2, ... , 0.8} are

x =   0 : 0
x = 0.1 : 0.25728658641487911
x = 0.2 : 0.44654239804174406
x = 0.3 : 0.59632253897119802
x = 0.4 : 0.72028629569299074
x = 0.5 : 0.82603834828984657
x = 0.6 : 0.91825063385856021
x = 0.7 : 1
x = 0.8 : 1

Parameters

the argument of the CDF

the first parameter of the distribution (strictly less than b)

the second parameter of the distribution

the third parameter of the distribution (strictly positive)

Returns

the CDF of the Bradford distribution

References

John Burkardt's library of statistical C++ routines, http://www.csit.fsu.edu/~burkardt/cpp_src/prob/prob.html

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Result –