I was thinking about the work I did when I worked in biostatistics at MD Anderson. This work was practical rather than mathematically elegant, useful in its time but not of long-term interest. However, one result came out of this work that I would call elegant, and that was a symmetry I found.

Let *X* be a beta(*a*, *b*) random variable and let *Y* be a beta(*c*, *d*) random variable. Let *g*(*a*, *b*, *c*, *d*) be the probability that a sample from *X* is larger than a sample from *Y*.

*g*(*a*, *b*, *c*, *d*) = Prob(*X* > *Y*)

This function often appeared in the inner loop of a simulation and so we spent thousands of CPU-hours computing its values. I looked for ways to evaluate this function more quickly, and along the way I found a symmetry.

The function I call *g* was studied by W. R. Thompson in 1933 [1]. Thompson noted two symmetries:

*g*(*a*, *b*, *c*, *d*) = 1 − *g*(*c*, *d*, *a*, *b*)

and

*g*(*a*, *b*, *c*, *d*) = *g*(*d*, *c*, *b*, *a*)

I found an additional symmetry:

*g*(*a*, *b*, *c*, *d*) = *g*(*d*, *b*, *c*, *a*).

The only reference to this result in a journal article as far as I know is a paper I wrote with Saralees Nadarajah [2]. That paper cites an MD Anderson technical report which is no longer online, but I saved a copy here.

## Related posts

- Random inequalities I: Introduction
- Random inequalities VI: Gamma distributions
- Twenty weeks down to twenty minutes

[1] W. R. Thompson. On the Likelihood that One Unknown Probability Exceeds Another in View of the Evidence of Two Samples. *Biometrika*, Volume 25, Issue 4. pp. 285 – 294.

[2] John D. Cook and Saralees Nadarajah. Stochastic Inequality Probabilities for Adaptively Randomized Clinical Trials. Biometrical Journal. 48 (2006) pp 256–365.