The previous post looked at an application of the beta-binomial distribution. The probability mass function for a beta-binomial with parameters *n*, *a*, and *b* is given by

The mean μ and the variance σ² are given by

Solving for *a* and *b* to meet a specified mean and variance appears at first to require solving a cubic equation, but it’s easier than that.

If we define *p* = *a*/(*a*+*b*) then the system becomes

We assume *n* is known and so *p* is known, and we are left with a linear equation for *a*. With a little work we find

I verified the calculations above with Mathematica.

One use for solving for *a* and *b *would be to fit a beta-binomial distribution to data using moment matching. I don’t know how robust this would be, but at least it’s something.

Another application would be to find a beta-binomial prior distribution with elicited mean and variance.