LogNormal distribution
Story
If \(\ln y\) is Normally distributed, then y is LogNormally distributed.
Example
A measure of fold change in gene expression can be LogNormally distributed.
Parameters
As for the Normal distribution, there are two parameters, the location parameter \(\mu\) and the scale parameter \(\sigma\). Note that \(\mu\) is the mean of \(\ln y\), not of \(y\) itself. That is, \(\langle\ln y\rangle_{\mathrm{LogNorm}} = \mu\). Similarly, \(\langle(\ln y  \mu)^2\rangle_{\mathrm{LogNorm}} = \sigma^2\).
Support
The LogNormal distribution is supported on the set of positive real numbers.
Probability density function
Moments
Mean: \(\displaystyle{\mathrm{e}^{\mu + \sigma^2/2}}\)
Variance: \(\left(\mathrm{e}^{\sigma^2}  1\right)\mathrm{e}^{2\mu + \sigma^2}\)
Usage
Package 
Syntax 

NumPy 

SciPy 

Stan 

Notes
Be careful not to get confused. The LogNormal distribution describes the distribution of \(y\) given that \(\ln y\) is Normally distributed. It does not describe the distribution of \(\ln y\).
The way location, scale, and shape parameters work in SciPy for the LogNormal distribution is confusing. If you want to specify a LogNormal distribution as we have defined it using
scipy.stats
, use a shape parameter equal to \(\sigma\), a location parameter of zero, and a scale parameter given by \(\mathrm{e}^\mu\). For example, to compute the PDF, you would usescipy.stats.lognorm.pdf(y, sigma, loc=0, scale=np.exp(mu))
.