Let X be a beta distributed random variable with parameters $\alpha$ and $\beta$, variance V and skew S.
Is it possible that several different sets of $\alpha$ and $\beta$ have the same variance and skew? Intuitively, I expect the answer to be "No"
How can I find the set of parameters $\alpha$ and $\beta$ that yield that distribution to have a variance of V and a skew of S?
The solution should come from resolving the following system of equation but for some reason I failed to do so.
$$V = \frac{\alpha \beta}{(\alpha + \beta)^2(\alpha+\beta+1)}$$ $$S = \frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha + \beta + 2)\sqrt{\alpha \beta}}$$