Beta function¶
It is defined as the following integral. $\( B(a,b) = \int_0^1 \theta^{\alpha - 1}(1-\theta)^{\beta - 1}d\theta \)$
Gama function¶
\(B(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\)
Because of this it can be related to the Binomial coefficient:
\[
B(a,b) = \frac{(a-1)!(b-1)!}{(a+b-1)!} = \frac{a+b}{ab}\frac{1}{\binom{a+b}{a}}
\]