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Cauchy Distribution


CauchyDistributionFigure

The Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution, is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.

Let theta represent the angle that a line, with fixed point of rotation, makes with the vertical axis, as shown above. Then

tantheta=x/b
(1)
theta=tan^(-1)(x/b)
(2)
dtheta=1/(1+(x^2)/(b^2))(dx)/b
(3)
=(bdx)/(b^2+x^2),
(4)

so the distribution of angle theta is given by

 (dtheta)/pi=1/pi(bdx)/(b^2+x^2).
(5)

This is normalized over all angles, since

 int_(-pi/2)^(pi/2)(dtheta)/pi=1
(6)

and

int_(-infty)^infty1/pi(bdx)/(b^2+x^2)=1/pi[tan^(-1)(x/b)]_(-infty)^infty
(7)
=1/pi[1/2pi-(-1/2pi)]
(8)
=1.
(9)
CauchyDistribution

The general Cauchy distribution and its cumulative distribution can be written as

P(x)=1/pib/((x-m)^2+b^2)
(10)
D(x)=1/2+1/pitan^(-1)((x-m)/b),
(11)

where b is the half width at half maximum and m is the statistical median. In the illustration about, m=0.

The Cauchy distribution is implemented in the Wolfram Language as CauchyDistribution[m, Gamma/2].

The characteristic function is

phi(t)=1/piint_(-infty)^inftye^(itx)(1/2Gamma)/((1/2Gamma)^2+(x-m)^2)dx
(12)
=e^(imt-Gamma|t|/2).
(13)

The moments mu_n of the distribution are undefined since the integrals

 mu_n=int_(-infty)^inftyGamma/(2pi)(x^n)/((x-m)^2+(1/2Gamma)^2)dx
(14)

diverge for n>=1.

If X and Y are variates with a normal distribution, then Z=X/Y has a Cauchy distribution with statistical median m=0 and full width

 Gamma=(2sigma_x)/(sigma_y).
(15)

The sum of n variates each from a Cauchy distribution has itself a Cauchy distribution, as can be seen from

P_n(x)=F_t^(-1){[phi(t)]^n}(x)
(16)
=((1/2nGamma))/(pi[(1/2nGamma)^2+(x-nm)^2]),
(17)

where phi(t) is the characteristic function and F_t^(-1)[f(t)](x) is the inverse Fourier transform, taken with parameters a=b=1.


See also

Normal Distribution

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References

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 104, 1984.Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 114-115, 1992.

Referenced on Wolfram|Alpha

Cauchy Distribution

Cite this as:

Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html

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