과제6

f_{X_i}(x_i;\mu) = {1\over2\pi}~ \exp \bigg( -{(x_i-\mu)^2 \over 2} \bigg)~I(\mu >0)

\begin{align*} &L(\mu) = \prod_{i=1}^{n}~ f_{X_i}(x_i;\mu) \\[20pt] &= \bigg( {1\over2\pi} \bigg)^n \exp \bigg( -{1 \over 2}~ \sum~ (x_i-\mu)^2\bigg)~I(\mu > 0) \\[30pt] &l(\mu) = \log(L(\mu)) \\[20pt] &= -{n \over 2}~\log(2\pi) -{1 \over 2}~\sum_{i=1}^{n} ~(x_i - \mu)^2~I(\mu > 0) \end{align*}

{\partial \over \partial\mu}~l(\mu) = \sum_{i=1}^{n} ~(x_i - \mu)\cdot(-1) = \sum_{i=1}^{n} ~(x_i - \mu) = 0 \\[20pt] \sum_{i=1}^{n}~x_i = n\mu, \kern{10pt} {1 \over n}\sum_{i=1}^{n}~x_i = \overline{X} = \mu