Log Normal (Cobb-Douglass) Distribution#

Has one shape parameter $\sigma$ >0. (Notice that the “Regress” $A=\log S$ where $S$ is the scale parameter and $A$ is the mean of the underlying normal distribution). The support is $x\geq0$ .

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

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

Notice that using JKB notation we have $\theta=L,$ $\zeta=\log S$ and we have given the so-called antilognormal form of the distribution. This is more consistent with the location, scale parameter description of general probability distributions.

$h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right].$

Also, note that if $X$ is a log-normally distributed random-variable with $L=0$ and $S$ and shape parameter $\sigma.$ Then, $\log X$ is normally distributed with variance $\sigma^{2}$ and mean $\log S.$

Implementation: scipy.stats.lognorm