Beta Distribution

Continuous distribution.

• $\alpha > 0$,
  • parameter
• $\beta > 0$,
  • parameter
• $\mathrm{\Gamma}(\alpha)$,
  • gamma function
• $\mathrm{B}(\alpha, \beta)$,
  • beta function

\[\mathrm{B}(\alpha, \beta) := \frac{ \Gamma(\alpha) \Gamma(\beta) }{ \Gamma(\alpha + \beta) },\]

$\mathrm{Beta}(\alpha, \beta)$ is the p.d.f of Beta distribution given $\alpha$ and $\beta$.

\[x \in [0, 1], \ \mathrm{Beta}(x; \alpha, \beta) := \frac{ x^{\alpha-1} (1 - x)^{\beta - 1} }{ \mathrm{B}(\alpha, \beta) } .\]

Reference

• Bernoulli distribution - Wikipedia