Cauchy Distribution

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 segmenttilted 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 | | | (1) | | | | (2) | | | | (3) | | | | (4) | so the distribution of angle is given by | (5) |
This is normalized over all angles, since | (6) | and | | | (7) | | | | (8) | | | | (9) |

The general Cauchy distribution and its cumulative distribution can be written as | | | (10) | | | | (11) | 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 | | | (12) | | | | (13) |
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 | | | (16) | | | | (17) | where is the characteristic function and is the inverse Fourier transform, taken with parameters .

Definition | | Properties | The distribution is symmetric about the parameter a.The parameter b determines the width of the distributionThe central moments are undefined for Cauchy distributed data, which is a consequence of the long tails of the density function (see example below).The median and the mode are equal to the location parameter a.The standard Cauchy