Cauchy(a,b) - no Crystal Ball distribution, =Student(1)*b+a
The standard Cauchy distribution can be derived from the ratio of two independent Normal distributions, i.e. if X and Y are two independent Normal(0,1) distributions, then
X/Y = Cauchy(0,1)
The Cauchy(a,b) is shifted to have a median at a, and to have b times the spread of a Cauchy(0,1). Examples of the Cauchy distribution are given below:
The Cauchy distribution is not often used in risk analysis. It is used in mechanical and electrical theory, physical anthropology and measurement and calibration problems. For example, in physics it is usually called a Lorentzian distribution, where it is the distribution of the energy of an unstable state in quantum mechanics. It is also used to model the points of impact of a fixed straight line of particles emitted from a point source.
The most common use of a Cauchy distribution is to show how 'smart' you are by quoting it whenever someone generalizes about how distributions are used, because it is the exception in many ways: in principle, it has no defined mean (though by symmetry this is usually accepted as being its median = a), and no other defined moments.
The distribution is symmetric about a and the spread of the distribution increases with increasing b. The Cauchy distribution is peculiar and most noted because none of its moments are well defined (i.e. mean, standard deviation, etc.), their determination being the difference between two integrals that both sum to infinity. Although it looks similar to the Normal distribution, it has much heavier tails. From X/Y = Cauchy(0,1) above you'll appreciate that the reciprocal of a Cauchy distribution is another Cauchy distribution (it is just swapping the two Normal distributions around). The range a - b to a + b contains 50% of the probability area.
Pronounced Co-shee, it takes its name from Augustin Louis Cauchy after a paper he wrote on the distribution in 1853, but Pierre de Fermat was the first to have written about the Cauchy distribution in the mid 17th century (now, is that fair?). The Cauchy distribution is actually a shifted, rescaled Student-t distribution with one degree of freedom: the Crystal Ball distribution "Student(0,1,1)*b+a" will generate a Cauchy(a, b) distribution with Latin Hypercube sampling.
The Cauchy distribution is also sometimes called the Lorentz distribution.