doc/source/tutorial/stats/continuous_lognorm.rst
.. _continuous-lognorm:
Has 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).
The support is :math:x\geq0.
.. math:: :nowrap:
\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*}
.. math:: :nowrap:
\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 :math:\theta=L, :math:\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.
.. math::
h\left[X\right]=\frac{1}{2}\left[1+\log\left(2\pi\right)+2\log\left(\sigma\right)\right].
Also, 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.
Implementation: scipy.stats.lognorm