Cauchy Distribution
The Cauchy distribution, also called the Lorentzian Distribution, describes resonance behavior. It also describes thedistribution of horizontal distances at which a Line Segment tilted at a random Angle cuts thex-Axis. Let 
represent the Angle that a line, with fixed point of rotation, makeswith 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 withMean and full width
 | (16) |
References
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 114-115, 1992.
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