Is there a closed form of the regularized incomplete Beta function $I_x(\alpha, \beta)$ when $\alpha + \beta = 1$?
A nice closed form based on arcsin exists for the case where $\alpha = \beta = 1/2$ (see here). Is there a similar one for the more general case?
My motivation is that I'd like to find, given $p \in [0, 1]$, the $\alpha$ such that the $\Pr(X > 0.5) = p$ where $X \sim Beta(\alpha, 1- \alpha)$. I would like to do this efficiently for any $p \in [0, 1]$ – which should be easy with the closed form of the regularized incomplete Beta function.