Hallo,
die Log-Likelihood-Funktion lautet
\( l(\lambda) = \sum_{i=1}^n \log(\lambda \exp(-\lambda x_i)) \).
Es ist
\( \frac{d l}{d \lambda} = \sum_{i=1}^n \frac{\exp(-\lambda x_i) - \lambda x_i \exp(-\lambda x_i)}{\lambda \exp(-\lambda x_i)} \)
\( = \sum_{i=1}^n \left( \frac{1}{\lambda} - x_i \right) \)
\( = \frac{n}{\lambda} - \sum_{i=1}^n x_i = 0 \).
Daraus folgt
\( \lambda = \frac{1}{\frac{1}{n} \sum_{i=1}^n x_i} \)
\( = \frac{1}{\overline{x}} \).
Grüße
Mister
