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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 μ,ϕ\mu, \phi are required.

    μ>0,ϕ>0\mu>0,\phi>0

  • Continuous distribution defined on entire range

  • This distribution is always symmetric.

  • Mean, standard deviation, skewness and kurtosis are not defined.

Probability​

  • Cumulative distribution function

    F(x)=12+1πtan⁡−1x−μϕF(x)=\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\frac{x-\mu}{\phi}

  • Probability density function

    f(x)=1πϕ[1+(x−μϕ)2]f(x)=\frac{1}{\pi\phi\left[1+\left(\frac{x-\mu}{\phi}\right)^2\right]}

  • How to compute these on Excel.

AB
1DataDescription
20.5Value for which you want the distribution
38Value of parameter Mu
42Value of parameter Phi
5FormulaDescription (Result)
6=0.5+ATAN((A2-A3)/A4)/PI()Cumulative distribution function for the terms above
7=1/(PI()*A4*(1+((A2-A3)/A4)^2))Probability density function for the terms above

Quantile​

  • Inverse function of cumulative distribution function

    F−1(P)=μ+ϕ[tan⁡π(P−12)]F^{-1}(P)=\mu+\phi\left[\tan\pi\left(P-\frac{1}{2}\right)\right]

  • How to compute this on Excel.

AB
1DataDescription
20.7Probability associated with the distribution
31.7Value of parameter Mu
40.9Value of parameter Phi
5FormulaDescription (Result)
6=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,

    x=μ+ϕ[tan⁡π(U−12)]x=\mu+\phi\left[\tan\pi\left(U-\frac{1}{2}\right)\right]

  • How to generate random numbers on Excel.

AB
1DataDescription
20.5Value of parameter Mu
30.5Value of parameter Phi
4FormulaDescription (Result)
5=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​