This is documentation for an old release of SciPy (version 0.17.0). Read this page Search for this page in the documentation of the latest stable release (version 1.15.1).

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 standard form is $x>0$

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

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

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