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
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so the distribution of angle is given by
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This is normalized over all angles, since
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and
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The general Cauchy distribution and its cumulative distribution can be written as
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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
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The moments of the distribution are undefined since the integrals
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diverge for .
If and are variates with a normal distribution, then has a Cauchy distribution with statistical median and full width
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The sum of variates each from a Cauchy distribution has itself a Cauchy distribution, as can be seen from
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where is the characteristic function and is the inverse Fourier transform, taken with parameters .