疲劳寿命(Birnbaum-Saunders)分布¶
该分布的pdf是逆高斯分布的平均值 \(\left(\mu=1\right)\) 和倒数逆高斯pdf \(\left(\mu=1\right)\) 。我们在这里的jkb符号后面加上 \(\beta=S.\) 有一个形状参数 \(c>0\) ，并且支持是 \(x\geq0\) 。
\begin{eqnarray*} f\left(x;c\right) & = & \frac{x+1}{2c\sqrt{2\pi x^{3}}}\exp\left(-\frac{\left(x-1\right)^{2}}{2xc^{2}}\right)\\
F\left(x;c\right) & = & \Phi\left(\frac{1}{c}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\right)\\
G\left(q;c\right) & = & \frac{1}{4}\left[c\Phi^{-1}\left(q\right)+\sqrt{c^{2}\left(\Phi^{-1}\left(q\right)\right)^{2}+4}\right]^{2}\end{eqnarray*}
\[M\left(t\right)=c\sqrt{2\pi}\exp\left(\frac{1}{c^{2}}\left(1-\sqrt{1-2c^{2}t}\right)\right)\Left(1+\FRAC{1}{\SQRT{1-2c^{2}t}}\Right)\]
\BEGIN{eqnarray*}\mu&=&\frac{c^{2}}{2}+1\\
\mu{2}&=&c^{2}\left(\frac{5}{4}c^{2}+1\right)\\
\Gamma_{1}&=&\frac{4c\sqrt{11c^{2}+6}}{\left(5c^{2}+4\right)^{3/2}}\\
\Gamma_{2}&=&\frac{6c^{2}\left(93c^{2}+41\right)}{\left(5c^{2}+4\right)^{2}}\end{eqnarray*}