Beta distribution

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.