1 2.. _continuous-lognorm: 3 4Log Normal (Cobb-Douglass) Distribution 5======================================= 6 7Has one shape parameter :math:`\sigma` >0. (Notice that the "Regress" :math:`A=\log S` where :math:`S` is the scale parameter and :math:`A` is the mean of the underlying normal distribution). 8The support is :math:`x\geq0`. 9 10.. math:: 11 :nowrap: 12 13 \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)\\ 14 F\left(x;\sigma\right) & = & \Phi\left(\frac{\log x}{\sigma}\right)\\ 15 G\left(q;\sigma\right) & = & \exp\left( \sigma\Phi^{-1}\left(q\right)\right) \end{eqnarray*} 16 17.. math:: 18 :nowrap: 19 20 \begin{eqnarray*} \mu & = & \exp\left(\sigma^{2}/2\right)\\ 21 \mu_{2} & = & \exp\left(\sigma^{2}\right)\left[\exp\left(\sigma^{2}\right)-1\right]\\ 22 \gamma_{1} & = & \sqrt{p-1}\left(2+p\right)\\ 23 \gamma_{2} & = & p^{4}+2p^{3}+3p^{2}-6\quad\quad p=e^{\sigma^{2}}\end{eqnarray*} 24 25Notice that using JKB notation we have :math:`\theta=L,` :math:`\zeta=\log S` and we have given the so-called antilognormal form of the 26distribution. This is more consistent with the location, scale 27parameter description of general probability distributions. 28 29.. math:: 30 31 h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right]. 32 33Also, note that if :math:`X` is a log-normally distributed random-variable with :math:`L=0` and :math:`S` and shape parameter :math:`\sigma.` Then, :math:`\log X` is normally distributed with variance :math:`\sigma^{2}` and mean :math:`\log S.` 34 35Implementation: `scipy.stats.lognorm` 36