author: yit date: 2013/12/30
\[ \begin{aligned} p(y|λ) &= \frac{λ^y\exp(-λ)}{y!} \\ \end{aligned} \]
\[ \begin{aligned} L(λ) &= \Pi_{i} {p(y_{i}|λ)} \\ &= \Pi_{i} \frac{λ^{y_{i}}\exp(-λ)}{y_{i}!} \end{aligned} \]
\[ \begin{aligned} \log L(λ) &= \log{\Pi_{i} \frac{λ^{y_{i}}\exp(-λ)}{y_{i}!}} \\ &= \sum_{i} \log{\frac{λ^{y_{i}}\exp(-λ)}{y_{i}!}} \\ &= \sum_{i} \log{λ^{y_{i}}} + \sum_{i} \log{\exp(-λ)} + \sum_{i} \log{\frac{1}{y_{i}!}} \\ &= \sum_{i} y_{i} \log{λ} + \sum_{i} -λ \log{e} - \sum_{i} \log{{y_{i}!}} \\ &= \sum_{i} y_{i} \log{λ} + \sum_{i} -λ - \sum_{i} \sum_{k}^{i}\log{{y_{k}}} \\ &= \sum_{i} ( y_{i} \log{λ} -λ - \sum_{k}^{i}\log{{y_{k}}} ) \\ \end{aligned} \]