One advantage to the conventional definition of the Beta function $B(s,t)$ is that a random variable whose probability distribution is the Beta distribution with probability distribution proportional to $x^{s-1}(1-x)^{t-1}\ dx$ has expected value $s/(s+t)$.

One advantage to the conventional definition of the Beta function $B(s,t)$ is that a random variable whose probability distribution is the Beta distribution with probability distribution proportional to $x^{s-1}(1-x)^{t-1}\ dx$ for $0\le x\le 1$ has expected value $s/(s+t)$.