Two weeks ago I wrote a series of posts on random inequalities: part I, part II, part III. In the process of writing these, I found an error in a tech report I wrote five years ago. I’ve posted a corrected version and describe the changes here.

Suppose X_{1} is a Cauchy random variable with median m_{1} and scale s_{1} and similarly for X_{2}. Then X_{1} – X_{2} is a Cauchy random variable with median m_{1} – m_{2} and scale s_{1} + s_{2}. Then P(X_{1} > X_{2}) equals

P(X_{1} – X_{2} > 0) = P(m_{1} – m_{2} + (s_{1} + s_{2}) C > 0)

where C is a Cauchy random variable with median 0 and scale 1. This reduces to

P(C < (m_{1} – m_{2})/(s_{1} + s_{2})) = 1/2 + atan( (m_{1} – m_{2})/(s_{1} + s_{2}) )/π.

The original version was missing the factor of 1/2. This is obviously wrong because it would say that P(X_{1} > X_{2}) is negative when m_{1} < m_{2}.

By the way, I was told in college that the Cauchy distribution is an impractical curiosity, something more useful for developing counterexamples than modeling real phenomena. That was an overstatement. Thick-tailed distributions like the Cauchy often arise in applications, sometimes directly (see Noise, The Black Swan) or indirectly (for example, robust or default prior distributions).

**Update**: See part V on beta distributions.

The Cauchy distribution’s impracticality is definitely an overstatement. It commonly arises in many areas of physics, such as in the spectral representation of quantum field theories.

I think the ratio of two Gaussians is also Cauchy distributed. So if Gaussians are considered frequent, then the ratio of them shouldn’t be considered “impractical curiosity”

> and scale s1 + s2

(s1 + s2) or sqrt(s1*s1 + s2*s2) ? Or is “scale” already proportional of variance (squared stddev)?

If X has finite variance, Var(aX) = a^2 Var(X). So variance is proportional to scale squared.

It’s really unexpectedly to me, that scale-parameters are added, not scale-squared %)

It’s very interesting, that there can be other law for scale summation, if variance is not existed. Thanks for the surprise

Whoever told you in college that the Cauchy distribution is not useful to model real phenomena has no idea of physics.

The Cauchy distribution in Physics is called the Lorentzian, and it appears as the energy distribution of a metastable quantum state (a stable state would have a sharp, zero-variance degenerate distribution). The reason is that the fourier transform of a Cauchy distribution is an exponential, a metastable state can be modelled as having an exponentially decaying survival probability, and time and energy are conjugate variables related by a form of Heisenberg’s uncertainty principle.