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27.2 Incomplete Beta

The incomplete beta function, $\text{B}(x; a, b)$, is defined for $x \in [0, 1]$ and $a, b \geq 0$ such that $a + b \neq 0$ by $$\text{B}(x; \, a, b) \ = \ \int_0^x u^{a - 1} \, (1 - u)^{b - 1} \, du,$$ where $\text{B}(a, b)$ is the beta function defined in appendix. If $x = 1$, the incomplete beta function reduces to the beta function, $\text{B}(1; a, b) = \text{B}(a, b)$.

The regularized incomplete beta function divides the incomplete beta function by the beta function, $$I_x(a, b) \ = \ \frac{\text{B}(x; \, a, b)}{B(a, b)} \, .$$