Log-normal distribution

Taking the exponential of a Normal random variable with mean \(\lambda\) and standard deviation \(\zeta\) gives a log-normal random variable; i.e., the natural logarithm of the random variable follows a Normal distribution.

In the following, let \(X\) be a random variable that follows a \emph{log-normal} distribution with parameters \(\lambda\) and \(\zeta\), and let \(x\) denote a particular outcome of \(X\). If the distribution of \(X\) is shifted by parameter \(\varepsilon\), the distribution is referred to as shifted log-normal distribution (If not mentioned explicitly, this parameter equals zero).

import fesslix as flx
flx.load_engine()

import numpy as np
import fesslix.plot as flx_plot
import matplotlib.pyplot as plt
%matplotlib inline

Random Number Generator: MT19937 - initialized with rand()=1222282124;
Random Number Generator: MT19937 - initialized with 1000 initial calls.

Syntax

property logn

Log-normal distribution

logn is a distribution type (flx_rv_type) for Random variables in Fesslix.

Parametrization:

Parameters of the distribution can be specified as additional key-value pairs in an object of type flx_rv_config. The following combinations of parameters are accepted:

• mu, sd

• lambda, zeta

• mode, sd

• median, sd

• median, cov

• mode, cov

• val_1, pr_1, val_2, pr_2

• cov, val_1, pr_1

Additionally, the key-value pair epsilon is allowed for any parametrization.

The interpretation of the parameters is:

• mu (flxParaPosNo0): mean value; only positive values larger than epsilon are allowed

• sd (flxParaPosNo0): standard deviation

• epsilon (flxParaPos0, default: 0.0): right-shift of distribution

• mode (flxParaPosNo0): mode of distribution; only positive values larger than epsilon are allowed

• median (flxParaPosNo0): median of distribution; only positive values larger than epsilon are allowed

• cov (flxParaPosNo0): coefficient of variation. If epsilon is set, the cov is interpreted as cov=sd/(mu-epsilon).

• lambda (flxPara): mean value of the underlying Normal distribution

• zeta (flxParaPosNo0): standard deviation of the underlying Normal distribution

• val_1 (flxParaPosNo0): pr_1 quantile

• `pr_1 (flxParaPr): probability that the value of the distribution is smaller or equal than val_1

• val_2 (flxParaPosNo0): pr_2 quantile

• `pr_2 (flxParaPr): probability that the value of the distribution is smaller or equal than val_2

Example:

rv_1 = flx.rv({'name':'rv_1', 'type':'logn', 'mu':2., 'sd':5. })
rv_2 = flx.rv({'name':'rv_2', 'type':'logn', 'lambda':0., 'zeta':1. })
rv_3 = flx.rv({'name':'rv_3', 'type':'logn', 'mode':2., 'sd':5. })
rv_4 = flx.rv({'name':'rv_4', 'type':'logn', 'median':2., 'sd':5. })
rv_5 = flx.rv({'name':'rv_5', 'type':'logn', 'mode':2., 'cov':1. })
rv_6 = flx.rv({'name':'rv_6', 'type':'logn', 'median':2., 'cov':1. })
rv_7 = flx.rv({'name':'rv_7', 'type':'logn', 'cov':0.1, 'val_1':2.5, 'pr_1':0.05 })
rv_8 = flx.rv({'name':'rv_8', 'type':'logn', 'val_1':2.5, 'pr_1':0.05, 'val_2':3.2, 'pr_2':0.5 })
rv_lst = [ rv_1, rv_2, rv_3, rv_4, rv_5, rv_6, rv_7, rv_8 ]

Properties

Notation

\(X\sim\mathrm{ln}\mathcal{N}(\lambda,\zeta,\varepsilon) = \exp\left[\mathcal{N}\left(\lambda,\zeta\right)\right]+\varepsilon\)

Parameters

\(\lambda\in\mathbb{R}\), \(\zeta\in(0,\infty)\), \(\varepsilon\in\mathbb{R}\)

Support

\(x\in(\varepsilon,\infty)\)

Mean

\(\mu_X = \exp\left(\lambda+\frac{\zeta^2}{2}\right) + \varepsilon\)

Standard deviation

\(\sigma_X = \sqrt{\exp\left(\zeta^2\right)-1} \cdot \exp\left(\lambda+\frac{\zeta^2}{2}\right)\)

Median

\(\exp\left(\lambda\right) + \varepsilon\)

Mode

\(\exp\left(\lambda-\zeta^2\right) + \varepsilon\)

Skewness

\(\left( \exp\left(\zeta^2\right)+2 \right) \cdot \sqrt{\exp\left(\zeta^2\right)-1}\)

Excess kurtosis

\(\exp\left(4\zeta^2\right) + 2\exp\left(3\zeta^2\right) + 3\exp\left(2\zeta^2\right) - 6\)

Entropy

\(\frac{1}{2} + \frac{1}{2}\ln\left(2\pi \zeta^2\right) + \lambda\)

PDF

The PDF \(f_X(x)\) of the log-normal distribution is:

\[f_X(x) = \frac{1}{(x-\varepsilon)\cdot\zeta\cdot\sqrt{2\pi}} \cdot\exp\left[-\frac{1}{2}\left(\frac{\ln(x-\varepsilon)-\lambda}{\zeta}\right)^2\right]\]

The PDF \(f_X(x)\) can also be expressed in terms of the PDF of the Standard Normal distribution \(\varphi(\cdot)\):

\[f_X(x) = \frac{1}{(x-\varepsilon)\cdot\zeta}\cdot\varphi\left(\frac{\ln(x-\varepsilon)-\lambda}{\zeta}\right)\]

x_up = 8.

fig, ax = plt.subplots(figsize=(10, 4))

for rv in rv_lst:
    flx_plot.draw_pdf(ax, rv, config_dict={'label':rv.get_name(), 'x_low':1e-6, 'x_up':x_up}) 

ax.set_ylim([0., None])
ax.set_xlim([0., x_up])
plt.xlabel(r"$x$")
plt.ylabel(r"$f_X(x)$")
plt.legend()
plt.show()

CDF

The CDF \(F_X(x)\) of the log-normal distribution is defined as:

\[F_X(x) = \Phi\left(\frac{\ln(x-\varepsilon)-\lambda}{\zeta}\right)\]

where \(\Phi(\cdot)\) is the CDF of the Standard Normal distribution.

x_up = 8.

fig, ax = plt.subplots(figsize=(10, 4))

for rv in rv_lst:
    flx_plot.draw_cdf(ax, rv, config_dict={'label':rv.get_name(), 'x_low':1e-6, 'x_up':x_up}) 
    
ax.set_ylim([0., 1.])
ax.set_xlim([0., x_up])
plt.xlabel(r"$x$")
plt.ylabel(r"$F_X(x)$")
plt.legend()
plt.show()

Quantile function

The quantile function of the log-normal distribution is:

\[F_X^{-1}(p) = \exp\left( \lambda + \zeta \cdot\Phi^{-1}(p) \right) + \varepsilon \;, \quad p\in(0,1)\]

where \(\Phi^{-1}(\cdot)\) is the quantile function of the Standard Normal distribution.

Standardizing Normal random variables

The log-normal random variable \(X\) can be transformed to a Standard Normal distribution \(U\) through:

\[U = \frac{\ln(X-\varepsilon)-\lambda}{\zeta}\]

Conversely, a log-normal random variable \(X\) with parameters \(\lambda\) and \(\zeta\) can be generated from a standard Normal variable as:

\[X = \exp\left(\lambda + \zeta \cdot U\right)+\varepsilon\]

Parametrization in terms of \(\mu_X\) and \(\sigma_X\)

If the mean and standard deviation are given, the parameters \(\lambda\) and \(\zeta\) can be derived as:

\[\lambda = \ln\left(\mu_X-\varepsilon\right)-\frac{1}{2}\ln\left(\frac{\sigma_X^2}{(\mu_X-\varepsilon)^2}+1\right)\]

\[\zeta = \sqrt{\ln\left(\frac{\sigma_X^2}{(\mu_X-\varepsilon)^2}+1\right)}\]

Covariance of two correlated log-normal random variables

The correlation coefficient \(\rho\) between two correlated log-normal random variables \(X_1\) and \(X_2\) is

\[\rho = \frac{\operatorname{COV}(X_1,X_2)}{\sigma_1\sigma_2} = \frac{\exp\left(\rho^\prime\sqrt{\ln(\delta_1^2+1)\ln(\delta_2^2+1)}\right)-1}{\delta_1\delta_2}\]

where \(\delta_1\) and \(\delta_2\) is the coefficient of variation of \(X_1\) and \(X_2\), respectively; \(\rho^\prime\) is the correlation coefficient of the underlying standard Normal random variables.