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## 26.2 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, $I_x(a, b) \ = \ \frac{\text{B}(x; \, a, b)}{B(a, b)} \, .$