Beta Distribution

The beta distribution is a continuous probability distribution defined on the interval \(\left[0, 1\right]\), characterized by two shape parameters, and is often used to model random variables representing proportions or probabilities.

A beta random variable is denoted as followed:

\[ X \sim \text{Beta}(\alpha, \beta) \]

Beta Function

The beta distribution is based on the beta function:

\[ B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} dx \]

Probability Distribution Function (PDF)

The pdf for the beta distribution is given as followed:

\[\begin{split} f_X(x) = \begin{cases} \frac{1}{B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1}, & \text{if } x \in (0,1) \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Expectation

The expected value of the beta distribution can be derived as followed:

\[\begin{split} \begin{align} EX &= \int_0^1 x f_X(x) dx \\ &= \frac{1}{B(\alpha, \beta)} \int_0^1 x^\alpha (1-x)^{\beta-1} dx \\ &= \frac{1}{B(\alpha, \beta)} \int_0^1 x^{(\alpha+1)-1} (1-x)^{\beta-1} dx \\ &= \frac{B(\alpha+1,\beta)}{B(\alpha, \beta)} \\ &= \frac{\Gamma(\alpha+1) \Gamma(\beta) \Gamma(\alpha + \beta)}{\Gamma(\alpha + \beta + 1) \Gamma(\alpha) \Gamma(\beta)} \\ &= \frac{\alpha \Gamma(\alpha) \Gamma(\alpha + \beta)}{(\alpha + \beta) \Gamma(\alpha + \beta) \Gamma(\alpha)} \\ &= \frac{\alpha}{\alpha + \beta} \\ \end{align} \end{split}\]

This can be generalized to solve for any moment:

\[\begin{split} \begin{align} EX^r &= \int_0^1 x^r f_X(x) dx \\ &= \frac{1}{B(\alpha, \beta)} \int_0^1 x^{\alpha - 1 + r} (1-x)^{\beta-1} dx \\ &= \frac{1}{B(\alpha, \beta)} \int_0^1 x^{(\alpha+r)-1} (1-x)^{\beta-1} dx \\ &= \frac{B(\alpha+r,\beta)}{B(\alpha, \beta)} \\ &= \frac{\Gamma(\alpha+r) \Gamma(\beta) \Gamma(\alpha + \beta)}{\Gamma(\alpha + \beta + r) \Gamma(\alpha) \Gamma(\beta)} \\ &= \frac{\Gamma(\alpha+r) \Gamma(\alpha + \beta)}{\Gamma(\alpha + \beta + r) \Gamma(\alpha)} \\ \end{align} \end{split}\]

Variance

Using our generalized moment formula, we can easily find \(Var(X)\):

\[ Var(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \]