# The Beta Distribution

Posted by Beetle B. on Tue 06 June 2017

The beta distribution

The parameters are $$\alpha,\beta>0$$, and $$A,B$$ with $$B\ge A$$.

\begin{equation*} f(x;\alpha,\beta,A,B)=\frac{1}{B-A}\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\left(\frac{x-A}{B-A}\right)^{\alpha-1}\left(\frac{B-x}{B-A}\right)^{\beta-1} \end{equation*}

For $$A\le x\le B$$

When $$A=0,B=1$$, it is called the standard Beta distribution.

\begin{equation*} \mu=A+(B-A)\frac{\alpha}{\alpha+\beta} \end{equation*}
\begin{equation*} \sigma^{2}=\frac{(B-A)^{2}\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)} \end{equation*}

Essentially, $$A,B$$ are used to stretch/shift the curve. It is defined only for values between $$A$$ and $$B$$.

It is often used to model the proportion of percentage of a quantity in different samples (e.g. % of 24-hour day an individual is asleep).