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 X1 is a Cauchy random variable with median m1 and scale s1 and similarly for X2. Then X1 – X2 is a Cauchy random variable with median m1 – m2 and scale s1 + s2. Then P(X1 > X2) equals
P(X1 – X2 > 0) = P(m1 – m2 + (s1 + s2) C > 0)
where C is a Cauchy random variable with median 0 and scale 1. This reduces to
P(C < (m1 – m2)/(s1 + s2)) = 1/2 + atan( (m1 – m2)/(s1 + s2) )/π.
The original version was missing the factor of 1/2. This is obviously wrong because it would say that P(X1 > X2) is negative when m1 < m2.
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.