## Cauchy Distribution

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)

This is normalized over all angles, since
 (5)

and
 (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)

See also Gaussian Distribution, Normal Distribution

References

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

© 1996-9 Eric W. Weisstein
1999-05-26