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.