Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $I_p(a,b)$ is the regularized incomplete beta function.

Question: Given $\alpha,\beta,$ and $c>0$, what is $p$?

Attempt: I need to invert the CDF's, but the problem is the arguments of the CDFs are not the same. I'm not sure how to proceed.


Using the inductive property



It is obtained:


From here there is no general solution in a closed form $\forall\,\alpha$,$\beta$ but last equation is very easy to solve numerically


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