Log-Normal Distribution#

The log-normal distribution is a two-parameter continuous probability distribution. A log-normal random variable is a variable whose (natural) logarithm is a normally distributed random variable. The table below summarizes some important aspects of the distribution.

Notation

\(X \sim \mathcal{N}_{\mathrm{log}} (\mu, \sigma)\)

Parameters

\(\mu \in \mathbb{R}\) (scale parameter)

\(\sigma > 0\) (shape parameter)

Support

\(\mathcal{D}_X = (0, \infty)\)

PDF

\(f_X (x; \mu, \sigma) = \begin{cases} 0 & x \leq 0 \\ \frac{1}{\sigma \sqrt{2 \pi}} \exp{\left[ - \frac{1}{2} \left(\frac{\log(x) - \mu}{\sigma} \right)^2 \right]} & x > 0 \end{cases}\)

CDF

\(F_X (x; \mu, \sigma) = \begin{cases} 0 & x \leq 0 \\ \frac{1}{2} \left[ 1 + \mathrm{erf}\left( \frac{\log(x) - \mu}{\sigma \sqrt{2}}\right) \right] & x > 0 \end{cases}\)

ICDF

\(F^{-1}_X (x) = \exp{\left[ \mu + \sqrt{2} \, \sigma \, \mathrm{erf}^{-1}(2 x - 1) \right]}\)

Note

There are various ways to parameterize the log-normal distribution. In the parameterization adopted here, \(\mu\) and \(\sigma\) correspond to the mean and standard deviation of the underlying normal distribution, respectively. Under SciPy parameterization, \(e^\mu\) corresponds to the scale parameter, while \(\sigma\) corresponds to the shape parameter.

The plots of probability density function (PDF), cumulative distribution function (CDF), as well as the histogram of a sample (\(5000\) points) for different parameter values are shown below.

../../_images/lognormal_1_0.png