분포별 가능도함수를 정리해보자. (가능도함수 = 확률분포의 n제곱)
정규
\begin{align*}
f_{X_i}(x; \theta) &= \dfrac{1}{\sqrt{2\pi\sigma}}
\exp \bigg(- \dfrac{(x-\mu)^2}{2\sigma^2} \bigg)
\\[20pt]
\prod_{i=1}^{ n}
~f_{x_i}(x; \theta) &=
\bigg({1\over \sqrt{2\pi\sigma}}\bigg)^n
~\exp \bigg(-{1 \over 2\sigma^2}~
\sum_{i=1}^{n}~
\big(X_i - \mu\big)^2
\bigg)
\end{align*}
\theta = \mu인 경우
\begin{align*}
l(\theta) &=
-{n \over 2}~log(2\pi\sigma^2)
-{1 \over 2\sigma^2}~\sum_{i=1}^{n}
\big(X_i - \mu\big)^2
\\[20pt]
\dfrac{d}{d\theta}~l(\theta) &=
-{1 \over \sigma^2}~
\sum_{i=1}^{n}\big(X_i - \mu\big)
\cdot (-1)
= {1 \over \sigma^2}~
\big(n\overline{X} - n\mu\big)
\\[20pt]
\dfrac{d^2}{d^2\theta}~l(\theta) &=-{n \over \sigma^2}
\end{align*}
\theta = \sigma^2인 경우
\begin{align*}
l(\theta) &=
-{n \over 2}~log(2\pi\sigma^2)
-{1 \over 2\sigma^2}~\sum_{i=1}^{n}
\big(X_i - \mu\big)^2
\\[20pt]
\dfrac{d}{d\theta}~l(\theta)
&=-{n \over \sigma}-
{1 \over 2}\cdot\bigg(
-{2\sigma\over\sigma^4}\bigg)
~\sum_{i=1}^{n}
\big(X_i - \mu\big)^2
\\[20pt]
&= -{n \over \sigma}
+ {1 \over \sigma^3}
~\sum_{i=1}^{n}
\big(X_i - \mu\big)^2
\\[20pt]
\dfrac{d^2}{d^2\theta}~l(\theta) &= -{n \over \sigma^2}~
+ {3 \over \sigma^4}
~\sum_{i=1}^{n}
\big(X_i - \mu\big)^2
\end{align*}
기하
\begin{align*}
f_X(x) &= p(1-p)^{n-1}
\\[15pt]
\prod_{i=1}^{n}~f_{X_i}(x) &=
p^n~(1-p)^{\sum x_i -n}
\end{align*}
감마
\begin{align*}
f_{X_i}(x_i) &=
{1\over\Gamma(\alpha)~\beta^\alpha}
~x^{\alpha-1}
~e^{-x/\beta}
\\[20pt]
\prod_{i=1}^{n}~f_{X_i}(x_i) &=
\end{align*}
l(\theta) =
지수
f_{X_i}(x_i) = \lambda e^{\lambda x}
\prod_{i=1}^{n}~f_{X_i}(x_i) =
포아송
f_{X_i}(x_i) =
{e^{-\lambda}~\lambda^x \over x!}
\prod_{i=1}^{n}~f_{X_i}(x_i) =
{e^{-\lambda}~\lambda^x \over x!}