Beta Distribution
Definition
A random variable $X \sim \text{Beta}(\alpha, \beta)$ with $\alpha, \beta > 0$ has PDF
$$ f(x; \alpha, \beta) = \frac{x^{\alpha - 1}(1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \quad x \in [0, 1], $$where $B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$ is the Beta function.
Moments
$$ \mathbb{E}[X] = \frac{\alpha}{\alpha + \beta}, \qquad \text{Var}(X) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}. $$Role in Bayesian Inference
The Beta distribution is the conjugate prior for the Bernoulli and Binomial likelihoods.