====== 와이블분포 (Weibull Distribution) ======
===== 표기 =====
 $\alpha$ : [[분포]]의 척도, [[척도모수]](Scale Parameter)

 $\beta$ : [[분포]]의 형상, [[형상모수]](Shape Parameter)

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

  * $$ \alpha \in ( \ 0 \ , \ \infty \ ) $$
  * $$ \beta \in ( \ 0 \ , \ \infty \ ) $$
===== 받침 =====
 $$ x \in [ \ 0 \ , \ \infty \ ) $$
===== 확률밀도함수 =====
 $$ f(x) =  \frac{\beta}{\alpha} \left( \frac{x}{\alpha} \right)^{\beta - 1} \cdot \exp \left[ - \left( \frac{x}{\alpha} \right)^{\beta} \right] $$

<plot>
 set title "Weibull Distribution PDF"
 set size 0.7
 set xrange [0:10]
 set yrange [0:1.2]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 f(x,a,b) = (b/a)*((x/a)**(b-1))*exp(-((x/a)**b))

 plot f(x,1,0.5) title "Wei(1,0.5)", \
  f(x,1,1) title "Wei(1,1)", \
  f(x,1,2) title "Wei(1,2)", \
  f(x,3,0.5) title "Wei(2,0.5)"
</plot>
===== [누적분포함수] =====
 $$ F(x) = 1 - \exp \left[ - \left( \frac{x}{\alpha} \right)^{\beta} \right] $$

<plot>
 set title "Weibull Distribution CDF"
 set size 0.7
 set xrange [0:10]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"

 f(x,a,b) = 1-exp(-((x/a)**b))

 plot f(x,1,0.5) title "Wei(1,0.5)", \
  f(x,1,1) title "Wei(1,1)", \
  f(x,1,2) title "Wei(1,2)", \
  f(x,2,0.5) title "Wei(3,0.5)"
</plot>
===== 기대값 =====
 $$ E(X) = \alpha \cdot \Gamma \left(1+\frac{1}{\beta} \right) $$
===== 분산 =====
 $$ Var(X) = \alpha^{2} \left[ \Gamma \left( 1+\frac{2}{\beta} \right) - \Gamma^{2} \left( 1+\frac{1}{\beta} \right) \right] $$
===== 왜도 =====
 $$ \gamma_{1} = \frac{2 \Gamma^{3} (1 + \beta^{-1}) - 3 \Gamma (1 + \beta^{-1}) \Gamma (1 + 2 \beta^{-1}) + \Gamma (1 + 3 \beta^{-1})}{ \left[ \Gamma (1 + 2 \beta^{-1}) - \Gamma^{2} (1 + \beta^{-1}) \right]^{3/2} } $$
===== 첨도 =====
 $$ \gamma_{2} = \frac{12 \Gamma^{2} (1 + \beta^{-1}) \Gamma (1 + 2 \beta^{-1}) - 3 \Gamma^{2} (1 + 2 \beta^{-1}) - 4 \Gamma (1 + \beta^{-1}) \Gamma (1 + 3 \beta^{-1}) + \Gamma (1 + 4 \beta^{-1}) -6 \Gamma^{4} (1 + \beta^{-1})}{ \left[ \Gamma (1 + 2 \beta^{-1}) - \Gamma^{2} (1 + \beta^{-1}) \right]^{2} } $$
===== 원적률 =====
 $$ \mu'_{1} = \alpha \Gamma (1 + \beta^{-1}) $$

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

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

 $$ \mu'_{4} = \alpha^{4} \Gamma (1 + 4 \beta^{-1}) $$
===== 고장률함수 =====
 $$ \lambda(x) = \frac{\beta}{\alpha} \left( \frac{x}{\alpha} \right)^{\beta - 1} $$

----
  * [[분포]]
  * [[연속형 분포]]