CodeCogs

Evaluates the Angle distribution CDF.

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Interface

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

using namespace Statistics::Distributions::Continuous::Angle;

inline double CDF(double x, int n)

This component evaluates the CDF of the Angle distribution with given arguments. x is an angle between 0 and $\pi$ , corresponding to the angle made in an n dimensional space, between a fixed line passing through the origin, and an arbitrary line that also passes through the origin, which is specified by choosing any point on the n dimensional sphere with uniform probability.

The Angle CDF is defined by the following formula

$\mathrm{D}(x) = \frac{\Gamma\left(\frac{n}{2}\right)} {\sqrt{\pi} \,\Gamma\left(\frac{n-1}{2}\right)} \int_0^x \sin^{n-2}(t) \mathrm{d}t \qquad \mbox{if } n > 2$

(1)

and

$\mathrm{D}(x) = \frac{x}{\pi} \qquad \mbox{if } n = 2$

(2)

for all

$x \in [0, \pi]$

(3)

The following figure illustrates the typical shape of the Angle distribution for a range of n: \graph x=0:3.5 n=2:6:5

Example 1

In the given example, the CDF is evaluated using a fixed value for n, equal to 4, while the first argument x takes values from 0 up to 2 with a step equal to 0.2. The maximum number of precision digits, implicitly set to 17, may be changed through the <em> PRECISION </em> define.

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

#define PRECISION 17

int main()
{
  std::cout << "The values of the Angle CDF with n = 4 and";
  std::cout << std::endl;
  std::cout << "x = {0, 0.2, 0.4, ..., 1.8, 2} are" << std::endl;
  std::cout << std::endl;
  for (double x = 0; x < 2.1; x += 0.2)
  {
    std::cout << std::setprecision(2);
    std::cout << "x = " << std::setw(3) << x << " : ";
    std::cout << std::setprecision(PRECISION);
    std::cout << Stats::Dists::Continuous::Angle::CDF(x, 4);
    std::cout << std::endl;
  }
  return 0;
}

Output

The values of the Angle CDF with n = 4 and
x = {0, 0.2, 0.4, ..., 1.8, 2} are

x =   0 : 0
x = 0.2 : 0.001684123127684655
x = 0.4 : 0.013153186649778228
x = 0.6 : 0.042647304023738355
x = 0.8 : 0.095560829038801032
x =   1 : 0.1735907059637424
x = 1.2 : 0.27446855935925973
x = 1.4 : 0.39231882068278456
x = 1.6 : 0.51858635136931963
x = 1.8 : 0.64338711110041569
x =   2 : 0.75706863044011896

Parameters

the argument of the CDF

the spatial dimension (must be at least 2)

Returns

the value of the Angle CDF evaluated with the given arguments

References

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

Interactive Calculator

Result –