The Cauchy distribution, also called the Lorentzian Distribution, describes 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

(1) | |||

(2) | |||

(3) |

so the distribution of Angle is given by

(4) |

(5) |

(6) |

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

(7) | |||

(8) |

where is the Full Width at Half Maximum ( in the above example) and is the Mean ( in the above example). The Characteristic Function is

(9) |

The Moments are given by

(10) | |||

(11) | |||

(12) |

and the Standard Deviation, Skewness, and Kurtosis by

(13) | |||

(14) | |||

(15) |

If and are variates with a Normal Distribution, then has a Cauchy distribution with
Mean and full width

(16) |

**References**

Spiegel, M. R. *Theory and Problems of Probability and Statistics.* New York: McGraw-Hill, pp. 114-115, 1992.

© 1996-9

1999-05-26