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