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