$$ L(\mu, S) = \sum_{n=1}^N \log P(x_n) = \sum_n \left( -\frac{1}{2} \log 2 \pi - \frac{1}{2} \log S - {(x_n - \mu)^2 \over 2 S} \right), \qquad \sigma^2 = S $$
$$ { \partial L \over \partial S} = \sum_n - \frac{1}{2S} - {(x_n - \mu)^2 \over 2} \cdot \left(- \frac{1}{S^2} \right) = \mathtt 0 \ \frac{N}{S} - \sum_n {(x_n - \mu)^2 \over S^2}, \quad /\cdot \frac{S^2}{N} \ S = \frac{1}{N} \sum_n (x_n - \mu)^2 $$
$$ L(\mu, S) = \sum_{n=1}^N \log P(x_n) = = \sum_{c=1}^C \log \pi_c^{m_c} = \sum_{c=1}^C \log \pi_c^{m_c} = \sum_{c = 1} m_c \log \pi_c $$
$$ {\partial L(\pi) \over \partial \pi_j } = \frac{m_j}{\pi_j} = 0 \implies \pi_j = \infty, \text{ale to nelze protože } \sum \pi_j = 1 $$
$$ \pi_c = { \alpha_c \over \sum_i \alpha_i } \ L = \sum m_c \log {\alpha_c \over \sum_i \alpha_i} = \sum m_c [\log \alpha_c - \log \sum_i \alpha_i] \ = \sum_c m_c \log \alpha_i - \sum_c m_c \log \sum_i \alpha_i $$
$$ {\partial L(\pi) \over \partial \pi_j } = \frac{m_j}{\alpha_j} - \sum_c m_c \frac{1}{\sum_i \alpha_j} = \mathtt 0 \ {\partial L(\pi) \over \partial \pi_j } = \frac{m_j}{\alpha_j} = {\sum_c m_c} \implies \pi_j = {\alpha_j \over \sum_i \alpha_i } = {m_j \over \sum_c m_c} $$