F比(或F)分布¶
的分布情况 \(\left(X_{{1}}/X_{{2}}\right)\left(\nu_{{2}}/\nu_{{1}}\right)\) 如果 \(X_{{1}}\) 与之呈卡方关系 \(v_{{1}}\) 自由度和 \(X_{{2}}\) 与之呈卡方关系 \(v_{{2}}\) 自由度。支持是 \(x\geq0\) 。
\BEGIN{eqnarray*}f\Left(x；\Nu_{1}，\Nu_{2}\Right)&=&\frac{\nu_{2}^{\nu_{2}/2}\nu_{1}^{\nu_{1}/2}x^{\nu_{1}/2-1}}{\left(\nu_{2}+\nu_{1}x\right)^{\left(\nu_{1}+\nu_{2}\right)/2}B\left(\frac{\nu_{1}}{2}，\frac{\nu_{2}}{2}\右)}\\
F\Left(x；v_{1}，v_{2}\Right)&=&I\Left(\frac{\nu_{1}x}{\nu_{2}+\nu_{1}x}；\frac{\nu_{1}}{2}，\frac{\nu_{2}}{2}\right)\\
g\Left(q；\nu_{1}，\nu_{2}\Right)&=&\Left(\frac{\nu_{2}}{i^{-1}\Left(q；\nu_{1}/2，\nu_{2}/2\right)}-\frac{\nu_{1}}{\nu_{2}}\right)^{-1}\\
\mU&=&\frac{\nu_{2}}{\nu_{2}-2}\quad\texpm{for}\nu_{2}>2\\
\MU_{2}&=&\frac{2\nu_{2}^{2}\left(\nu_{1}+\nu_{2}-2\right)}{\nu_{1}\left(\nu_{2}-2\right)^{2}\left(\nu_{2}-4\right)}\quad\textrm{for}v_{2}>4\\
\Gamma_{1}&=&\frac{2\left(2\nu_{1}+\nu_{2}-2\right)}{\nu_{2}-6}\sqrt{\frac{2\left(\nu_{2}-4\right)}{\nu_{1}\left(\nu_{1}+\nu_{2}-2\right)}}\quad\textrm{for}\nu_{2}>6\\
\Gamma_{2}&=&\frac{3\left(8+\left(\nu_{2}-6\right)\gamma_{1}^{2}\right)}{2\nu-16}\quad\textrm{for}\nu_{2}>8\end{等式*}
哪里 \(I\left(x;a,b\right)=I_{{x}}\left(a,b\right)\) 是正则化的不完全Beta函数。
实施： scipy.stats.f