Logistic (Sech 平方) 分布#
广义 Logistic 分布的一个特例，其中 \(c=1\)。支持范围为 \(x \in \mathbb{R}\).
该分布函数与其生存函数之间存在直接联系，通过 Fermi-Dirac 分布。即 scipy.stats.logistic.sf 等效于 Fermi-Dirac 分布。
\begin{eqnarray*} f\left(x\right) & = & \frac{\exp\left(-x\right)}{\left(1+\exp\left(-x\right)\right)^{2}}\\ F\left(x\right) & = & \frac{1}{1+\exp\left(-x\right)}\\ G\left(q\right) & = & -\log\left(1/q-1\right)\\ S\left(x\right) & = & n_F(x)=\frac{1}{1+\exp\left(x\right)}\end{eqnarray*}
\begin{eqnarray*} \mu & = & \gamma+\psi_{0}\left(1\right)=0\\ \mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(1\right)=\frac{\pi^{2}}{3}\\ \gamma_{1} & = & \frac{\psi_{2}\left(1\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}=0\\ \gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(1\right)\right)}{\mu_{2}^{2}}=\frac{6}{5}\\ m_{d} & = & \log1=0\\ m_{n} & = & -\log\left(2-1\right)=0\end{eqnarray*}
其中 \(\psi_m\) 是多伽马函数 \(\psi_m(z) = \frac{d^{m+1}}{dz^{m+1}} \log(\Gamma(z))\).
\[h\left[X\right]=1.\]