Beta distribution
In ExtDist: Extending the Range of Functions for Probability Distributions

========================================================

Probability density function:

$$f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathcal{B}(\alpha,\beta)}$$ with $\alpha$ and $\beta$ two shape parameters and $\mathcal B$ beta function.

Cumulative distribution function:

$$F(x) = \frac{\int_{0}^{x} y^{\alpha-1}(1-y)^{\beta-1}dy} {\mathcal{B}(\alpha,\beta)} =\mathcal{B}(x; \alpha,\beta)$$ with $\mathcal B (x; \alpha,\beta)$ incomplete beta function.

Log-likelihood function:

$$L(\alpha,\beta;X)=\sum_i\left[ (\alpha-1)\ln(x)+(\beta-1)\ln(1-x)-\ln \mathcal{B}(\alpha,\beta) \right]$$

Score function vector:

$$V(\mu,\sigma;X) =\left( \begin{array}{c} \frac{\partial L}{\partial \alpha} \ \frac{\partial L}{\partial \beta} \end{array} \right) =\sum_i \left( \begin{array}{c} \psi^{(0)}(\alpha+\beta)-\psi^{(0)}(\alpha)+\ln(x) \ \psi^{(0)}(\alpha+\beta)-\psi^{(0)}(\beta)+\ln(x) \end{array} \right) $$ with $\psi^{(0)}$ being log-gamma function.

Observed information matrix:

$$\mathcal J (\mu,\sigma;X)= \left( \begin{array}{cc} \psi^{(1)}(\alpha)-\psi^{(1)}(\alpha+\beta) & -\psi^{(1)}(\alpha+\beta) \ -\psi^{(1)}(\alpha+\beta) & \psi^{(1)}(\beta)-\psi^{(1)}(\alpha+\beta) \end{array} \right) $$ with $\psi^{(1)}$ being digamma function.