Beta-binomial with given mean and variance

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 and the variance ^2 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 fora. 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.

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