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 $\theta$ represent the Angle that a line, with fixed point of rotation, makes with the vertical axis, as shown above. Then

$$\tan\theta = {x\over b}$$ (1)

$$\theta = \tan^{-1}\left({x\over b}\right)$$ (2)

$$d\theta = - {1\over 1+{x^2\over b^2}} {dx\over b} = - {b\,dx\over b^2+x^2},$$ (3)

so the distribution of Angle $\theta$ is given by

$${d\theta\over\pi} = -{1\over\pi} {b\,dx\over b^2+x^2}.$$ (4)

This is normalized over all angles, since

$$\int_{-\pi/2}^{\pi/2} {d\theta\over \pi} = 1$$ (5)

and

$$-\int_{-\infty}^\infty {1\over \pi} {b\,dx\over b^2+x^2} = {1\over \pi}\left[{\tan^{-1}\left({b\over x}\right)}\right]_{-\infty}^\infty$$

$$= {1\over \pi} [{\textstyle{1\over 2}}\pi-(-{\textstyle{1\over 2}}\pi)] =1.$$ (6)

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

$$P(x) = {1\over\pi} {{\textstyle{1\over 2}}\Gamma\over (x-\mu)^2+({\textstyle{1\over 2}}\Gamma)^2}$$ (7)

$$D(x) = {1\over 2}+{1\over\pi}\tan^{-1}\left({x-\mu\over b}\right),$$ (8)

where $\Gamma$ is the Full Width at Half Maximum ($\Gamma=2b$ in the above example) and $\mu$ is the Mean ($\mu=0$ in the above example). The Characteristic Function is

$$\phi(t) = {1\over \pi}\int_{-\infty}^\infty {e^{it(\Gamma x/2-\mu)}\over 1+x^2}\,dx$$

$$= {e^{-i\mu t}\over\pi}\int_{-\infty}^\infty {\cos(\Gamma tx/2)\over 1+(\Gamma x/2)^2}\,dx$$

$$= e^{-i\mu t-\Gamma\vert t\vert/2}.$$ (9)

The Moments are given by

$$\mu_2 = \sigma^2 = \infty$$ (10)

$$\mu_3 = \left\{\begin{array}{ll} 0 & \mbox{for $\mu=0$}\\ \infty & \mbox{for $\mu\not=0$}\end{array}\right.$$ (11)

$$\mu_4 = \infty,$$ (12)

and the Standard Deviation, Skewness, and Kurtosis by

$$\sigma^2 = \infty$$ (13)

$$\gamma_1 = \left\{\begin{array}{ll} 0 & \mbox{for $\mu=0$}\\ \infty & \mbox{for $\mu\not=0$}\end{array}\right.$$ (14)

$$\gamma_2 = \infty.$$ (15)

If $X$ and $Y$ are variates with a Normal Distribution, then $Z\equiv X/Y$ has a Cauchy distribution with Mean $\mu=0$ and full width

$$\Gamma={2\sigma_y\over\sigma_x}.$$ (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.