The Beta distribution has the PDF:


for $0<x<1$, and $f(x)=0$ otherwise. The parameters $\alpha,\beta$ are positive real numbers.

The mean and variance are given by:


which can be inverted to give $\alpha,\beta$ in terms of the mean and the variance as $\alpha=\lambda\mu$ and $\beta=\lambda\left(1-\mu\right)$, where


Now I want to impose the condition that $\alpha,\beta \ge 1$. What does this imply for the mean and the variance? That is, is there a simple condition on $\mu,\sigma^2$ that is equivalent to $\alpha,\beta \ge 1$?


I was messing up the algebra, but I think I got it now:


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