## Quick Reference

A continuous random variable with probability density function f given by , where *k*>0 and *m* are parameters, is said to have a Cauchy distribution.

The graph of f is a bell-curve centred on *m*. The mode and the median are both equal to *m*, and the quartiles are *m*±*k*. A Cauchy distribution has no mean or variance, since, for example, does not exist.

The standard Cauchy distribution is given by *k*=1, *m*=0, and in this case the distribution is a *t*-distribution, with one degree of freedom.

Since the Cauchy distribution has neither a mean nor a variance, the central limit theorem does not apply. Instead, any linear combination of Cauchy variables has a Cauchy distribution (so that the mean of a random sample of observations from a Cauchy distribution has a Cauchy distribution).

If *X* and *Y* have independent standard normal distributions then *Y*/*X* has a standard Cauchy distribution. Equivalently, if *U* has a uniform continuous distribution on -½*π* < *u* < ½*π* then tan *U* has a standard Cauchy distribution. A geometrical representation of this is as follows. Let *O* be the origin of Cartesian coordinates, and let *A* be the point (0, 1). If the random point *P*, with coordinates (*X*, 0), is such that the angle *OAP* (=*U*, say) has a uniform continuous distribution on -½*π* < *u* < ½*π*, then *X* has a standard Cauchy distribution.

**Cauchy distribution.** The Cauchy distribution illustrated has *m* = 0 and *k* = 0.674. Also illustrated is the standard normal distribution. Both distributions have 25% of their area above 0.674 and 25% below-0.674. The fatter tails of the Cauchy distribution are apparent.

*Subjects:*
Probability and Statistics.

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