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 represent the angle that a line, with fixed point of rotation, makes with the vertical axis, as shown above. Then
so the distribution of angle is given by
(5)
This is normalized over all angles, since
(6)
and
The general Cauchy distribution and its cumulative distribution can be written as
where is the half width at half maximum and is the statistical median. In the illustration about, .
The Cauchy distribution is implemented in the Wolfram Language as CauchyDistribution[m, Gamma/2].
The characteristic function is
The moments of the distribution are undefined since the integrals
(14)
diverge for .
If and are variates with a normal distribution, then has a Cauchy distribution with statistical median and full width
(15)
The sum of variates each from a Cauchy distribution has itself a Cauchy distribution, as can be seen from
where is the characteristic function and is the inverse Fourier transform, taken with parameters .
See alsoNormal Distribution Explore with Wolfram|Alpha ReferencesPapoulis, 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|AlphaCauchy Distribution Cite this as:Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchyDistribution.html
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