die Log-Likelihood-Funktion ist
\( l(\beta) = \sum_{i=1}^{4} \log(f(x_i \mid \alpha, \beta)) \).
Ihre Ableitung hinsichtlich \( \beta \) ist
\( \frac{\partial l}{\partial \beta} = \sum_{i=1}^{4} \frac{\partial}{\partial \beta} \log(f(x_i \mid \alpha, \beta)) \)
\( = \sum_{i=1}^{4} \frac{\frac{\partial}{\partial \beta} f(x_i \mid \alpha, \beta)}{f(x_i \mid \alpha, \beta)} \)
\( = \sum_{i=1}^{4} \frac{\frac{1}{\Gamma(\alpha)} x_i^{\alpha - 1} \frac{\partial}{\partial \beta}\left( \beta^{\alpha} \exp(-\beta x_i) \right) }{\frac{1}{\Gamma(\alpha)} x_i^{\alpha - 1} \beta^{\alpha} \exp(-\beta x_i) } \)
\( = \sum_{i=1}^{4} \frac{\alpha \beta^{\alpha-1} \exp(-\beta x_i) - \beta^{\alpha} x_i \exp(-\beta x_i)}{\beta^{\alpha} \exp(-\beta x_i)} \)
\( = \sum_{i=1}^{4} \left( \frac{\alpha}{\beta} - x_i \right) \stackrel{!}{=} 0\).
Für \( \beta \) ergibt dies
\( \beta = \frac{4 \alpha}{\sum_{i=1}^{4} x_i} \left( = \frac{\alpha}{\bar{x}} \right) \)
\( = \frac{4 \cdot 1.41}{0.52 + 1.17 + 0.47 + 1.31} = 1.6254 \).
Mister
