Introduction to Cauchy distribution
The Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution, is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.
Let theta represent the angle that a line, with fixed point of rotation, makes with the vertical axis, as shown above. Then：
The general Cauchy distribution and its cumulative distribution can be written as
where b is the half width at half maximum and m is the statistical median. In the illustration about, m=0.
The Cauchy distribution is implemented in the Wolfram Language as CauchyDistribution[m, Gamma/2].
The characteristic function is
Source program for Cauchy distribution calculation
using System;
namespace Legalsoft.Truffer
{
/// <summary>
/// Cauchy distribution.
/// </summary>
public class Cauchydist
{
private double mu { get; set; }
private double sig { get; set; }
public Cauchydist(double mmu = 0.0, double ssig = 1.0)
{
this.mu = mmu;
this.sig = ssig;
if (sig <= 0.0)
{
throw new Exception("bad sig in Cauchydist");
}
}
public double p(double x)
{
return 0.318309886183790671 / (sig * (1.0 + Globals.SQR((x - mu) / sig)));
}
public double cdf(double x)
{
return 0.5 + 0.318309886183790671 * Math.Atan2(x - mu, sig);
}
public double invcdf(double p)
{
if (p <= 0.0 || p >= 1.0)
{
throw new Exception("bad p in Cauchydist");
}
return mu + sig * Math.Tan(3.14159265358979324 * (p - 0.5));
}
}
}