Beta distribution: trivial question

[markuz]

We all know that it is easy to get the beta mean and variance given the parameters α and β of the distribution (http://en.wikipedia.org/wiki/Beta_distribution).

Can we do right the opposite? I.e. is there any way to go uniquely from mean and variance to the parameters of the beta?

 

[mathman]

Based on the formulas in the link, the question can be reduced to solving a cubic equation in α or β as a function of the mean and variance. α and β are linear with respect to each other. Therefore there are at most three possible solutions. Since α and β are both required to be > 0, this could eliminate the extra solutions.

 

[economicsnerd]

Yes, there's a unique way. Convert the formula for mean into one for $\frac{\beta}{\alpha}$, then use that formula to simplify the variance formula into something linear in $\alpha$ and $\beta$. Then you'll have two straight-line relationships (one positively sloped, one negatively sloped) between the two parameters.