倒数高斯逆分布¶
从逆高斯(IG)中找到PDF， \(f_{{RIG}}\left(x;\mu\right)=\frac{{1}}{{x^{{2}}}}f_{{IG}}\left(\frac{{1}}{{x}};\mu\right)\) 定义为 \(x\geq0\) 作为
\BEGIN{eqnarray *}} f_{{IG}}\left(x;\mu\right) & = & \frac{{1}}{{\sqrt{{2\pi x^{{3}}}}}}\exp\left(-\frac{{\left(x-\mu\right)^{{2}}}}{{2x\mu^{{2}}}}\right).\\ F_{{IG}}\left(x;\mu\right) & = & \Phi\left(\frac{{1}}{{\sqrt{{x}}}}\frac{{x-\mu}}{{\mu}}\right)+\exp\left(\frac{{2}}{{\mu}}\right)\Phi\left(-\frac{{1}}{{\sqrt{{x}}}}\frac{{x+\mu}}{{\mu}}\right)\end{{eqnarray* }
\BEGIN{eqnarray *}} f_{{RIG}}\left(x;\mu\right) & = & \frac{{1}}{{\sqrt{{2\pi x}}}}\exp\left(-\frac{{\left(1-\mu x\right)^{{2}}}}{{2x\mu^{{2}}}}\right)\\ F_{{RIG}}\left(x;\mu\right) & = & 1-F_{{IG}}\left(\frac{{1}}{{x}},\mu\right)\\ & = & 1-\Phi\left(\frac{{1}}{{\sqrt{{x}}}}\frac{{1-\mu x}}{{\mu}}\right)-\exp\left(\frac{{2}}{{\mu}}\right)\Phi\left(-\frac{{1}}{{\sqrt{{x}}}}\frac{{1+\mu x}}{{\mu}}\right)\end{{eqnarray* }