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25.2 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, \[ I_x(a, b) \ = \ \frac{\text{B}(x; \, a, b)}{B(a, b)} \, . \]