# 逆高斯分布

參數 概率密度函數 ${\displaystyle \lambda >0}$ ${\displaystyle \mu >0}$ ${\displaystyle x\in (0,\infty )}$ ${\displaystyle \left[{\frac {\lambda }{2\pi x^{3}}}\right]^{1/2}\exp {\frac {-\lambda (x-\mu )^{2}}{2\mu ^{2}x}}}$ ${\displaystyle \Phi \left({\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}-1\right)\right)}$ ${\displaystyle +\exp \left({\frac {2\lambda }{\mu }}\right)\Phi \left(-{\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}+1\right)\right)}$ 其中${\displaystyle \Phi \left(\right)}$是高斯分布的累积分布函数 ${\displaystyle \mu }$ ${\displaystyle \mu \left[\left(1+{\frac {9\mu ^{2}}{4\lambda ^{2}}}\right)^{\frac {1}{2}}-{\frac {3\mu }{2\lambda }}\right]}$ ${\displaystyle {\frac {\mu ^{3}}{\lambda }}}$ ${\displaystyle 3\left({\frac {\mu }{\lambda }}\right)^{1/2}}$ ${\displaystyle 3+{\frac {15\mu }{\lambda }}}$ ${\displaystyle e^{\left({\frac {\lambda }{\mu }}\right)\left[1-{\sqrt {1-{\frac {2\mu ^{2}x}{\lambda }}}}\right]}}$

${\displaystyle f(x;\mu ,\lambda )=\left[{\frac {\lambda }{2\pi x^{3}}}\right]^{1/2}\exp {\frac {-\lambda (x-\mu )^{2}}{2\mu ^{2}x}}{\mbox{ for }}x>0.}$

Wald分布是μ = λ = 1时的逆高斯分布特例。

## 参考文献

• 逆高斯分布，作者Raj Chhikara与Leroy Folks
• 系统可靠性理论，作者Marvin Rausand与Arnljot Høyland