CauchyDistribution Class

[SerializableAttribute]
public class CauchyDistribution : ContinuousDistribution

<SerializableAttribute>
Public Class CauchyDistribution
	Inherits ContinuousDistribution

[SerializableAttribute]
public ref class CauchyDistribution : public ContinuousDistribution

[<SerializableAttribute>]
type CauchyDistribution = 
    class
        inherit ContinuousDistribution
    end

Inheritance

Object  →  Distribution  →  ContinuousDistribution  →  CauchyDistribution

The Cauchy distribution looks similar to the Normal distribution, but has a much heavier tail. It has some unusual properties. The mean is undefined, and the variance is infinite.

The Cauchy distribution is sometimes called the Lorentz distribution.

The scale parameter is a positive real number that determines how spread out the distribution is around the origin. Because the Cauchy has very heavy tails, it is often used in physics to describe resonance behavior. It can also be viewed as the ratio of two independent standard normal variables. This distribution can be more robust than the Normal distribution in contexts where outliers or large fluctuations occur more frequently.

This example shows how to create a CauchyDistribution object with a scale parameter of 2.5. It then demonstrates usage of several properties and methods of both CauchyDistribution and the ContinuousDistribution base class. The code uses top-level statements introduced in C# 9.

using System;
using Numerics.NET.Statistics.Distributions;

// Distribution parameters
var dist = new CauchyDistribution(2.5);
Console.WriteLine($"ScaleParameter: {dist.ScaleParameter}");

// Functions
double cdfValue = dist.DistributionFunction(1.2);
double pdfValue = dist.ProbabilityDensityFunction(1.2);
double invCdf = dist.InverseDistributionFunction(0.8);
Console.WriteLine($"Cdf(1.2) = {cdfValue}");
Console.WriteLine($"Pdf(1.2) = {pdfValue}");
Console.WriteLine($"InverseCdf(0.8) = {invCdf}");

// Moments
double mean = dist.Mean;
double median = dist.Median;
Console.WriteLine($"Mean = {mean}");
Console.WriteLine($"Median = {median}");

// Other properties
double sample = dist.Sample(new Random());
Console.WriteLine($"Sample = {sample}");

Reference