Logistic (Sech-squared) Distribution#
A special case of the Generalized Logistic distribution with \(c=1\). The support is \(x \in \mathbb{R}\).
This distribution function has a direct connection with the Fermi-Dirac
distribution via its survival function. I.e. scipy.stats.logistic.sf is
equivalent to the Fermi-Dirac distribution.
\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*}
where \(\psi_m\) is the polygamma function \(\psi_m(z) = \frac{d^{m+1}}{dz^{m+1}} \log(\Gamma(z))\).
\[h\left[X\right]=1.\]
Implementation: scipy.stats.logistic