> [!tldr] > A **beta distribution $\mathrm{Beta}(\alpha,\beta)$** has density (for $x \in [0,1]$) $f(x;\alpha, \beta)= \frac{1}{B(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}.$Here $B(\alpha,\beta)= \Gamma(\alpha)\Gamma(\beta) / \Gamma(\alpha + \beta)$ is the beta function. > > It can be used as a [[Bayesian Inference#Conjugation in Exponential Families|conjugate]] prior for probability-like parameters (e.g. $p$ in Bernoulli/binomial distributions). If $X \sim \mathrm{Beta}(\alpha, \beta)$, then it has moments $\mathbb{E}[X^{k}]=\frac{B(\alpha+k,\beta)}{B(\alpha,\beta)}=\frac{\Gamma(\alpha+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\alpha+k+\beta)}.$In particular, the first two moments are $\begin{align*} \mathbb{E}[X^{1}]&= \frac{\alpha! (\alpha+\beta-1)!}{(\alpha-1)!(\alpha+\beta)!}=\frac{\alpha}{\alpha+\beta},\\ \mathbb{E}[X^{2}]&= \frac{(\alpha+1)!(\alpha+\beta-1)!}{(\alpha-1)! (\alpha+\beta+1)!}=\frac{\alpha(\alpha+1)}{(\alpha+\beta+1)(\alpha+\beta)}. \end{align*}$ Therefore its variance is $\begin{align*} \mathrm{Var}(X)&= \mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}\\ &= \frac{\alpha(\alpha+1)(\alpha+\beta)-\alpha^{2}(\alpha+\beta+1)}{(\alpha+\beta+1)(\alpha+\beta)^{2}}\\ &= \frac{\alpha\beta}{(\alpha+\beta+1)(\alpha+\beta)^{2}}\\ &= \frac{\mu(1-\mu)}{\alpha+\beta+1}, \end{align*}$where $\mu=\mathbb{E}[X]$.