威布尔最小极值分布¶
一类有下界的极值分布。定义为 \(x>0\) 和 \(c>0\)
\BEGIN{eqnarray*}
F\Left(x；c\Right)&=&cx^{c-1}\exp\Left(-x^{c}\Right)\\
F\Left(x；c\Right)&=&1-\exp\Left(-x^{c}\Right)\\
g\Left(q；c\Right)&=&\Left [-\log\left(1-q\right)\right] ^{1/c}
\end{eqnarray*}
\[\mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right)\]
\BEGIN{eqnarray*}
\Mu&=&\Gamma\Left(1+\frac{1}{c}\Right)\\
\MU_{2}&=&\Gamma\Left(1+\frac{2}{c}\Right)-
\Gamma^{2}\Left(1+\frac{1}{c}\Right)\\
\Gamma_{1}&=&\frac{\Gamma\Left(1+\frac{3}{c}\right)-
3\Gamma\left(1+\frac{2}{c}\right)\Gamma\left(1+\frac{1}{c}\right)+
2\Gamma^{3}\Left(1+\frac{1}{c}\Right)}
{\mu{2}^{3/2}}\\
\Gamma_{2}&=&\frac{\Gamma\Left(1+\frac{4}{c}\right)-
4\Gamma\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{3}{c}\right)+
6\Gamma^{2}\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{2}{c}\right)-
3\Gamma^{4}\Left(1+\frac{1}{c}\Right)}
{\mu{2}^{2}}-3\\
M_{d}&=&\BEGIN{案例}
\Left(\frac{c-1}{c}\right)^{\frac{1}{c}}&\text{if}\；c>1\\
0&\text{if}\；c<=1
\结束{案例}\\
m_{n}&=&\ln\Left(2\Right)^{\frac{1}{c}}
\end{eqnarray*}
\[H\Left [X\right] =-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
哪里 \(\gamma\) 欧拉常数是否等于
\[\γ\约0.57721566490153286061。\]