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 -½π < u < ½π 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 -½π < u < ½π, 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.