Incomplete beta function

Summary

The incomplete beta function $B(x; a, b)$ is the integral of $t^{a-1}(1-t)^{b-1}$ from $0$ to $x$, for $0 \le x \le 1$. At $x = 1$ it is the beta function $B(a, b)$. Divided by $B(a, b)$ it becomes the regularised incomplete beta function $I_x(a, b)$, the distribution function of the beta distribution. It also gives the distribution functions of the binomial distribution and of Student's $t$-distribution.