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