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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

Probability

  • Cumulative distribution function
    F(x)=\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\frac{x-\mu}{\phi}
  • Probability density function
    f(x)=\frac{1}{\pi\phi\left[1+\left(\frac{x-\mu}{\phi}\right)^2\right]}
  • How to compute these on Excel.
     
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    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
    F^{-1}(P)=\mu+\phi\left[\tan\pi\left(P-\frac{1}{2}\right)\right]
  • How to compute this on Excel.
     
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    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,
    x=\mu+\phi\left[\tan\pi\left(U-\frac{1}{2}\right)\right]
  • How to generate random numbers on Excel.
     
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    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

 

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