The Beta Function comes up in the likelihood of the binomial distribution. Understanding its properties is useful for understanding the binomial distribution.

The beta function is given by $$B(a, b) = \int_0^1 p^{a-1}(1-p)^{b-1} \rm{d}p$$ for a and b positive. If you have $$N$$ flips of a coin of which $$k$$ turn heads the likelihood is proportional to $$p^{k}(1-p)^{N-k}$$ for the probability p between 0 and 1. So the beta function can be seen as the normaliser of the likelihood, with $$a = k + 1$$ and $$b = N - k + 1$$ (or inversely $$k = a - 1$$ and $$N = a + b - 2$$).

The integral can be evaluated directly when b is 1: $$B(a, 1) = \int_0^1 p^{a-1} = \frac{1}{a}$$.

Using Integration by Substitution of p with 1-p gives $$B(a, b) = B(b, a)$$. This makes sense because in the binomial distribution 0 and 1 are just labels, and if we switch the labels we should get the same overall normalisation.

Using Integration by Parts $$B(a, b+1) = \int_0^1 p^{a-1}(1-p)^{b} \rm{d}p = \Big[\frac{p^a(1-p)^b}{a}\Big]_0^1 + \frac{b}{a} \int_0^1 p^{a} (1-p)^{b-1}$$. The first term on the right hand side evaluates to 0 for all positive a and b giving $$B(a, b+1) = \frac{b}{a} B(a+1, b)$$.

This identity can be repeatedly applied to reduce b to 1 when it is a positive integer.

$B(a, m) = \frac{m-1}{a} B(a+1, m-1) = \frac{(m-1)(m-2)\cdots 1}{a(a+1)\cdots (a+m-1)} B(a+m-1, 1)$

And so this gives:

$B(a, m) = \frac{(m-1)!}{a(a+1)\cdots (a+m-1)(a+m)}$

Multiplying both sides by $$(a-1)!$$ gives:

$B(a, m) = \frac{(m-1)! (a-1)(a-2)\cdots 1}{(a+m)(a+m-1)\cdots 1}$

And so when a is also and integer this gives a simple form:

$B(n, m) = \frac{(m-1)! (n-1)!}{(n+m-1)!} = \frac{1}{(n+m-1) {n+m-2 \choose n-1}}$

Or in terms of the number of flips N of a coin and the number of positive results k:

$B(k+1, N-k+1) = \frac{1}{(N+1) {N \choose k}}$

The factorial can be generalised to all values with a positive real part using the Gamma Function, which can be seen with an appropriate change of variables:

$B(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

Many other useful features of the beta function can be obtained from this relation and $$\Gamma(a+1) = a \Gamma(a)$$. For example $$B(a+1, b) = \frac{a}{a+b} B(a, b)$$ and $$B(a+2, b) = \frac{a(a+1)}{(a+b)(a+b+1)} B(a, b)$$.

These will be useful when looking further into Binomials and the Beta distribution.