와이블분포 (Weibull Distribution)

$\alpha$ : 분포의 척도, 척도모수(Scale Parameter)

$\beta$ : 분포의 형상, 형상모수(Shape Parameter)

$$ X \sim Wei(\alpha , \beta)$$

• $$ \alpha \in ( \ 0 \ , \ \infty \ ) $$
• $$ \beta \in ( \ 0 \ , \ \infty \ ) $$

$$ x \in [ \ 0 \ , \ \infty \ ) $$

$$ f(x) = \frac{\beta}{\alpha} \left( \frac{x}{\alpha} \right)^{\beta - 1} \cdot \exp \left[ - \left( \frac{x}{\alpha} \right)^{\beta} \right] $$

$$ F(x) = 1 - \exp \left[ - \left( \frac{x}{\alpha} \right)^{\beta} \right] $$

$$ E(X) = \alpha \cdot \Gamma \left(1+\frac{1}{\beta} \right) $$

$$ Var(X) = \alpha^{2} \left[ \Gamma \left( 1+\frac{2}{\beta} \right) - \Gamma^{2} \left( 1+\frac{1}{\beta} \right) \right] $$

$$ \gamma_{1} = \frac{2 \Gamma^{3} (1 + \beta^{-1}) - 3 \Gamma (1 + \beta^{-1}) \Gamma (1 + 2 \beta^{-1}) + \Gamma (1 + 3 \beta^{-1})}{ \left[ \Gamma (1 + 2 \beta^{-1}) - \Gamma^{2} (1 + \beta^{-1}) \right]^{3/2} } $$

$$ \gamma_{2} = \frac{12 \Gamma^{2} (1 + \beta^{-1}) \Gamma (1 + 2 \beta^{-1}) - 3 \Gamma^{2} (1 + 2 \beta^{-1}) - 4 \Gamma (1 + \beta^{-1}) \Gamma (1 + 3 \beta^{-1}) + \Gamma (1 + 4 \beta^{-1}) -6 \Gamma^{4} (1 + \beta^{-1})}{ \left[ \Gamma (1 + 2 \beta^{-1}) - \Gamma^{2} (1 + \beta^{-1}) \right]^{2} } $$

$$ \mu'_{1} = \alpha \Gamma (1 + \beta^{-1}) $$

$$ \mu'_{2} = \alpha^{2} \Gamma (1 + 2 \beta^{-1}) $$

$$ \mu'_{3} = \alpha^{3} \Gamma (1 + 3 \beta^{-1}) $$

$$ \mu'_{4} = \alpha^{4} \Gamma (1 + 4 \beta^{-1}) $$

$$ \lambda(x) = \frac{\beta}{\alpha} \left( \frac{x}{\alpha} \right)^{\beta - 1} $$

• 분포
• 연속형 분포