Beta Distribution

The Beta density function is given as,

\[ \text{Beta}(r,s) \]

\[ f(x; r, s) = \frac{(r+s-1)!}{(r-1)!(s-1)!}x^{r-1}(1-x)^{s-1} \quad \text{for } r > 0, s > 0 \]

It is a important density function because many other density function are a version of the beta function. These density functions are said to be in the Beta family (e.g., Uniform).

Uniform Ordered Statistics

Consider a number line of \([0,1]\). At uniformly random we select \(n\) values from the number line. Let \(U_{(k)}\) be the random variable for the value of \(k\)th lowest value on the number line. In otherwords, \(U_{(k)}\) is the \(k\)th value in the number line when each of the \(n\) values are ordered.

It is found that \(U_{(k)}\) follows the beta distribution

\[ U_{(k)} \sim \text{Beta}(k, n-k+1) \]

Beta Integral

The beta integral is a type of integral in the form of,

\[ \int_0^1 x^{r-1}(1-x)^{s-1}~ dx= \frac{(r-1)!(s-1)!}{(r+s-1)!} \]

This derives the normalization factor for the beta distribution.