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) |
(4) |
(5) |
(6) |
The general Cauchy distribution and its cumulative distribution can be written as
(7) | |||
(8) |
(9) |
(10) | |||
(11) | |||
(12) |
(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