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The Regularized Incomplete Beta is expressed by $I_z(a,b) = \dfrac{\mathrm{B}(z;a,b)}{\mathrm{B}(a,b)} = \dfrac{\int_0^z u^{a-1}\left(1-u\right)^{b-1}\,du}{\int_0^1 u^{a-1}\left(1-u\right)^{b-1}\,du}$.

Is it possible to define each of the parameters $a, b$ individually as a closed-form expression in terms of the other variables?

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what do you mean by the words "individually ... in terms of the other variables"? Which "other"? You only have one variable $u$. –  Caran-d'Ache Aug 5 '13 at 6:09
Let's assume that $z$ and $b$ are constants and $y = I_z(a,b)$. Is it possible to express the parameter $a$ as a function of $y$? For example, calculating $y$ for different values of $a$ appears to be giving an almost exponential relationship. –  Kaln Aug 5 '13 at 9:07

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