- 확률에 대한 확률분포
- 주로 베이지안 추정에서의 사전분포로 활용한다.
f_X(x) =
{\Gamma(\alpha)~\Gamma(\beta)\over
\Gamma(\alpha+\beta)}~
x^{\alpha-1}~(1-x)^{\beta-1}
\\[15pt]
0 < x < 1,~\alpha > 0,~\beta > 0
정리
E(X) = {\alpha \over \alpha+\beta}
{\Gamma(\alpha)~\Gamma(\beta)\over
\Gamma(\alpha+\beta)}=
\int_{0}^{1}~
x^{\alpha-1}~(1-x)^{\beta-1}
적률
\begin{align*}
E(X) &= \int_{0}^{1}
x\cdot f_{X}(x)dx
\\[15pt]
&= \int_{0}^{1}~ {\Gamma(\alpha)~\Gamma(\beta)\over
\Gamma(\alpha+\beta)}~
x^{\alpha-1+1}~(1-x)^{\beta-1}
\\[15pt]
&={\Gamma(\alpha)~\Gamma(\beta)\over
\Gamma(\alpha+\beta)}~
\int_{0}^{1} x^{\alpha}~
(1-x)^{\beta-1}
\\[15pt]
&=
{\Gamma(\alpha)~\Gamma(\beta)\over
\Gamma(\alpha+\beta)} \cdot
{\Gamma(\alpha+1)~
\Gamma(\beta)\over
\Gamma(\alpha+1+\beta)}
\\[15pt]
&= {\alpha \over \alpha + \beta}
\end{align*}