PDF

Evaluates the Angle distribution PDF.

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Interface
PDF
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Interface

C++

PDF

doublePDF(	double	`x`
	int	`n`	)[inline]

This component evaluates the PDF of the Angle distribution with given arguments. x is an angle between 0 and $\inline \pi$ , 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

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

(2)

and

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

(3)

for all

$x \in [0, \pi]$

(4)

The mean of this distribution is given by

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

(5)

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.

There is an error with your graph parameters for PDF with options x=0:3 n=3

Error Message:Function PDF failed. Ensure that: Invalid C++

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

References

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

Parameters

x	the argument of the PDF
n	the spatial dimension (must be at least 2)

Returns

the value of the Angle PDF evaluated with the given arguments

Authors

Lucian Bentea (September 2005)

Source Code

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double	`PDF` (double x, int n)[inline] Evaluates the Angle distribution PDF.
double	`mean`