The Beta distribution is a continuous probability distribution defined on the interval $[0,1]$. It is parameterized by two shape parameters, $\alpha$ and $\beta$, which must satisfy $\alpha, \beta >0$. The distribution is commonly used to model probabilities or proportions. **PDF:** $ f_\Theta(\theta)=\frac{1}{B(\alpha, \beta)} * \theta^{\alpha-1}*(1-\theta)^{\beta-1}, \quad \text{where} \: 0 \le \theta \le1 $ where: - $\theta$ is the [[Random Variable]] - $\alpha, \beta$ are the shape parameters - $B(\alpha, \beta)$ is the Beta function, which serves as the normalization constant, to ensure that the [[Probability Density Function|PDF]] integrates to $1$. $B(\alpha, \beta)= \int_0^1 \theta^{\alpha-1} (1- \theta)^{\beta-1} d\theta$ The Beta function is expressed in terms of the [[Gamma Function]], or via factorials: $ B(\alpha, \beta)=\frac{\Gamma(\alpha)*\Gamma(\beta)}{\Gamma(\alpha+\beta)} = \frac{(\alpha-1) ! *(\beta-1) !}{(\alpha +\beta -1)!}$ **Expectation:** $ \mathbb E[\Theta]= \frac{\alpha}{\alpha+ \beta} $ **Variance:** $ \mathrm{Var}(\Theta)= \frac{\alpha \beta}{(\alpha+\beta)^2 *(\alpha+\beta+1)} $ **Mode:** $ \text{Mode}=\frac{\alpha-1}{\alpha+\beta-2}, \quad \text{for}\: \alpha>1, \, \beta>1$