# Cauchy distribution

## Where will you meet this distribution?

- Nuclear and Particle physics : Relativistic Breit-Wigner distribution
- Econophysics
- Special case of Levy distribution

## Shape of Distribution

### Basic Properties

- Two parameters are required.
- Continuous distribution defined on entire range
- This distribution is always symmetric.
- Mean, standard deviation, skewness and kurtosis are not defined.

### Probability

- Cumulative distribution function
- Probability density function
- How to compute these on Excel.

1 2 3 4 5 6 7 A B Data Description 0.5 Value for which you want the distribution 8 Value of parameter Mu 2 Value of parameter Phi Formula Description (Result) =0.5+ATAN((A2-A3)/A4)/PI() Cumulative distribution function for the terms above =1/(PI()*A4*(1+((A2-A3)/A4)^2)) Probability density function for the terms above

### Quantile

- Inverse function of cumulative distribution function
- How to compute this on Excel.

1 2 3 4 5 6

A B Data Description 0.7 Probability associated with the distribution 1.7 Value of parameter Mu 0.9 Value of parameter Phi Formula Description (Result) =A3+A4*(TAN(PI()*(A2-0.5))) Inverse of the cumulative distribution function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

Not defined

### Standard Deviation – How wide does the distribution spread? (Definition)

Not defined

### Skewness – Which side is the distribution distorted into? (Definition)

Not defined

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

Not defined

## Random Numbers

- Random number x is generated by inverse function method, which is for uniform random U,
- How to generate random numbers on Excel.

1 2 3 4 5

A B Data Description 0.5 Value of parameter Mu 0.5 Value of parameter Phi Formula Description (Result) =A2+A3*(TAN(PI()*(NTRAND(100)-0.5)) 100 Cauchy deviates based on Mersenne-Twister algorithm for which the parameters above Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

## NtRand Functions

Not supported yet

## Reference

- Wolfram Mathworld – Cauchy distribution
- Wikipedia – Cauchy distribution
- Statistics Online Computational Resource