The beta distribution has [[support]] from 0 to 1 and is highly flexible being defined by two parameters $\alpha$ and $\beta$.
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Beta_distribution_pdf.svg/1280px-Beta_distribution_pdf.svg.png" style="background-color:white;" />
# Notation
$X \sim Beta(\alpha, \beta)$
# Probability Density Function
$f(x) = \frac{1}{B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta - 1}$
where $B(\alpha, \beta)$ is the Beta function, a normalizing constant to ensure the total probability is $1$.
$B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)}$
where $\Gamma$ is the [[gamma function]].
# Expected Value
$E(X) = \frac{\alpha}{\alpha + \beta}$
# Variance
$Var(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$