Beta Distribution
Table of contents
Definition
The beta distribution is a continuous probability distribution with two parameters \(\alpha\) and \(\beta\).
\[p(x \mid \alpha, \beta) = B(x \mid \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}\]where
\[B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} = \int_0^1 x^{\alpha - 1} (1 - x)^{\beta - 1} dx\] \[\Gamma(\alpha) = \int_0^\infty t^{\alpha - 1} e^{-t} dt\] \[\Gamma(n) = (n - 1)!\] \[\text{mean: } \langle x \rangle = \frac{\alpha}{\alpha + \beta}\] \[\text{mode} = \frac{\alpha - 1}{\alpha + \beta - 2} \text{ (max of distribution)}\] \[\text{var}(x) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\]