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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 $\mu, \phi$ are required.
$\mu>0,\phi>0$
• Continuous distribution defined on entire range
• This distribution is always symmetric.
• Mean, standard deviation, skewness and kurtosis are not defined.

Probability

• Cumulative distribution function
$F(x)=\frac{1}{2}+\frac{1}{\pi}\tan^{-1}\frac{x-\mu}{\phi}$
• 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

Not defined

Not defined

Not defined

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