\(\newcommand{\R}{\mathbb{R}}\) \(\newcommand{\N}{\mathbb{N}}\) \(\newcommand{\E}{\mathbb{E}}\) \(\newcommand{\P}{\mathbb{P}}\) \(\newcommand{\var}{\text{var}}\) \(\newcommand{\sd}{\text{sd}}\) \(\newcommand{\cov}{\text{cov}}\) \(\newcommand{\cor}{\text{cor}}\) \(\newcommand{\skew}{\text{skew}}\) \(\newcommand{\kurt}{\text{kurt}}\)
  1. Virtual Laboratories
  2. 4. Special Distributions
  3. The Lognormal Distribution

The Lognormal Distribution

Basic Theory

Definition

Random variable \(X\) has the lognormal distribution with parameters \(\mu \in \R\) and \(\sigma \in (0, \infty)\) if \(\ln(X)\) has the normal distribution with mean \(\mu\) and standard deviation \(\sigma\).

Equivalently, \(X = e^{Y}\) where \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). The lognormal distribution is used to model continuous random quantities when the distribution is believed to be skewed, such as certain income and lifetime variables.

Distribution Functions

The probability density function of the lognormal distribution with parameters \(\mu\) and \(\sigma\) is given by \[ f(x) = \frac{1}{\sqrt{2 \pi} \sigma x} \exp \left(-\frac{\left[\ln(x) - \mu\right]^2}{2 \sigma^2} \right), \quad x \in (0, \infty) \]

  1. \( f \) increases and then decreases with mode at \( x = \exp\left(\mu - \sigma^2\right) \).
  2. \( f \) is concave upward then downward then upward again, with inflection points at \( x = \exp\left(\mu - \frac{3}{2} \sigma^2 \pm \frac{1}{2} \sigma \sqrt{\sigma^2 + 4}\right) \)
  3. \( f(x) \to 0 \) as \( x \downarrow 0 \) and as \( x \to \infty \).
Proof:

The form of the PDF follows from the change of variables theorem. The other parts follow from standard calculus.

In the special distribution simulator, select the lognormal distribution. Vary the parameters and note the shape and location of the probability density function. For selected values of the parameters, run the simulation 1000 times and compare the empirical density function to the true probability density function.

Let \(\Phi\) denote the standard normal distribution function. Recall that values of \(\Phi\) can be obtained from the special distribution calculator, as well as standard mathematical and statistical software packages. Thus, the following exercises show how to compute the lognormal distribution function and quantiles in terms of the standard normal distribution function and quantiles.

The lognormal distribution function \(F\) is given by \[ F(x) = \Phi \left[ \frac{\ln(x) - \mu}{\sigma} \right], \quad x \in (0, \infty) \]

The lognormal quantile function is given by \[ F^{-1}(p) = \exp\left[\mu + \sigma \Phi^{-1}(p)\right], \quad p \in (0, 1) \]

In the special distribution calculator, select the lognormal distribution. Vary the parameters and note the shape and location of the probability density function and the distribution function. With \(\mu = 0\) and \(\sigma = 1\), find the median and the first and third quartiles.

Moments

The moments of the lognormal distribution can be computed from the moment generating function of the normal distribution.

If \(X\) has the lognormal distribution with parameters \(\mu\) and \(\sigma\) then \[ \E(X^n) = \exp \left( n \mu + \frac{1}{2} n^2 \sigma^2 \right), \quad n \in \N \]

In particular, the mean and variance of \(X\) are

  1. \(\E(X) = \exp\left(\mu + \frac{1}{2} \sigma^2\right)\)
  2. \(\var(X) = \exp\left[2 (\mu + \sigma^2)\right] - \exp\left(2 \mu + \sigma^2\right)\)

Even though the lognormal distribution has finite moments of all orders, the moment generating function is infinite at any positive number. This property is one of the reasons for the fame of the lognormal distribution.

\(\E\left(e^{t X}\right) = \infty\) for every \(t \gt 0\).

In the simulation of the special distribution simulator, select the lognormal distribution. Vary the parameters and note the shape and location of the mean\( \pm \)standard deviation bar. For selected values of the parameters, run the simulation 1000 times and compare the empirical moments to the true moments.

Transformations

The most important transformations are the ones in the definition: if \(X\) has a lognormal distribution then \(\ln(X)\) has a normal distribution; conversely if \(Y\) has a normal distribution then \(e^X\) has a lognormal distribution.

For fixed \(\sigma\), the lognormal distribution with parameters \(\mu\) and \(\sigma\) is a scale family with scale parameter \(e^\mu\).

The lognormal distribution is a 2-parameter exponential family with natural parameters and natural statistics, respectively, given by

  1. \(\left( -\frac{1}{2 \sigma^2}, \frac{\mu}{\sigma^2} \right)\)
  2. \(\left(\ln^2(X), \ln(X)\right)\)

Computational Exercises

Suppose that the income \(X\) of a randomly chosen person in a certain population (in $1000 units) has the lognormal distribution with parameters \(\mu = 2\) and \(\sigma = 1\). Find \(\P(X \gt 20)\).

Answer:

\(\P(X \gt 20) = 0.1497\)

Suppose that the income \(X\) of a randomly chosen person in a certain population (in $1000 units) has the lognormal distribution with parameters \(\mu = 2\) and \(\sigma = 1\). Find each of the following:

  1. \(\E(X)\)
  2. \(\var(X)\)
Answer:
  1. \(\E(X) = e^{5/2} \approx 12.1825\)
  2. \(\sd(X) = \sqrt{e^6 - e^5} \approx 15.9629\)