CodeCogs

Evaluates the Angle distribution PDF.

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Interface

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

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

const double mean = 0.5*PI;

Overview

This component evaluates the PDF 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 PDF is defined by the following formula

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

(1)

and

$\mathrm{P}(x) = \frac{1}{\pi} \qquad \mbox{if }n = 2$

(2)

for all

$x \in [0, \pi]$

(3)

The mean of this distribution is given by

$\mu = \frac{\pi}{2} \qquad \forall n, \mbox{ the spatial dimension}$

(4)

In the example that follows, the PDF is evaluated using a fixed value for n, equal to 3, while the first argument x takes values from 0 up to 2 with a step equal to 0.2. The mean of the distribution, stored as a constant inside the same namespace, is also displayed. The maximum number of precision digits, implicitly set to 17, may be changed through the PRECISION define.

\graph x=0:3 n=3

Example 1

#include <codecogs/statistics/distributions/continuous/angle/pdf.h>
#include <iostream>
#include <iomanip>

#define PRECISION 17

int main()
{
  std::cout << "The values of the Angle PDF with n = 3 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::PDF(x, 3);
    std::cout << std::endl;
  }
  std::cout << std::endl;
  std::cout << "The mean of the Angle distribution is equal to";
  std::cout << std::endl;
  std::cout << Stats::Dists::Continuous::Angle::mean;
  std::cout << std::endl;
  return 0;
}

Output:

The values of the Angle PDF with n = 3 and
x = {0, 0.2, 0.4, ..., 1.8, 2} are

x = 0 : 0
x = 0.2 : 0.099334665397530608
x = 0.4 : 0.19470917115432523
x = 0.6 : 0.28232123669751769
x = 0.8 : 0.35867804544976134
x = 1 : 0.4207354924039482
x = 1.2 : 0.46601954298361309
x = 1.4 : 0.49272486499422996
x = 1.6 : 0.49978680152075255
x = 1.8 : 0.48692381543909757
x = 2 : 0.45464871341284085

The mean of the Angle distribution is equal to
1.5707963267948966

Parameters

the argument of the PDF

the spatial dimension (must be at least 2)

Returns

the value of the Angle PDF 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

DECLARATION