Beta Distribution

Parameters

• ${\mathbb{R}}_{>0}\ni \alpha$
• ${\mathbb{R}}_{>0}\ni \beta$

A random variable is said to be Beta distributed if it is described by the following

Probability Density Function

$$\begin{array}{rrcl}{p}_X(\cdot ,\alpha ,\beta ):& [0,1]& \to & \mathbb{R}\\ & x& \to & \{\begin{array}{rcl}\frac{1}{B(\alpha ,\beta )}{x}^{\alpha -1}(1-x{)}^{\beta -1},& \text{if}& x\in [0,1]\\ 0,& \text{if}& \text{otherwise},\end{array}\end{array}$$

where $B(\alpha ,\beta )=\Gamma (\alpha +\beta )/(\Gamma (\alpha )\Gamma (\beta ))$.

Moments

• Expected value: $E[X]=\alpha /(\alpha +\beta )$.
• Variance: $\text{Var}(X)=\alpha \beta /((\alpha +\beta {)}^2(\alpha +\beta +1))$.

The distribution is an appropriate model if the following assumptions are true:

Rationale

Let $X$ be a binomial random variable with parameters $n$ and $p$, where $p$ is unknown. By assuming that $p$ has a Beta distribution, then for $X=k$ (i.e., $k$ successes in $n$ Bernoulli trials) $X\sim \text{Beta}(\alpha +k,\beta +k)$.