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Gilbrat Distribution

Special case of the log-normal with $\sigma=1$ and $S=1.0$ (typically also $L=0.0$ )

$$\begin{eqnarray*} f\left(x;\sigma\right) & = & \frac{1}{x\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\log x\right)^{2}\right]\\ F\left(x;\sigma\right) & = & \Phi\left(\log x\right)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\log x}{\sqrt{2}}\right)\right]\\ G\left(q;\sigma\right) & = & \exp\left\{ \Phi^{-1}\left(q\right)\right\} \end{eqnarray*}$$

$$\begin{eqnarray*} \mu & = & \sqrt{e}\\ \mu_{2} & = & e\left[e-1\right]\\ \gamma_{1} & = & \sqrt{e-1}\left(2+e\right)\\ \gamma_{2} & = & e^{4}+2e^{3}+3e^{2}-6\end{eqnarray*}$$

$$\begin{eqnarray*} h\left[X\right] & = & \log\left(\sqrt{2\pi e}\right)\\ & \approx & 1.4189385332046727418\end{eqnarray*}$$

Implementation: scipy.stats.gilbrat