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Log-normal distribution

by , PhD

A random variable is said to have a log-normal distribution if its natural logarithm has a normal distribution.

In other words, the exponential of a normal random variable has a log-normal distribution.

Visual summary of the relation between the normal and log-normal distributions.

Table of Contents

Definition

Log-normal random variables are characterized as follows.

Definition Let X be a continuous random variable. Let its support be the set of strictly positive real numbers:[eq1]We say that X has a log-normal distribution with parameters mu and sigma^2 if its probability density function is[eq2]

The probability density function (pdf) of the log-normal distribution is plotted for different values of the volatility parameter sigma.

Relation to the normal distribution

The relation to the normal distribution is stated in the following proposition.

Proposition Let Y be a normal random variable with mean mu and variance sigma^2. Then the variable[eq3]has a log-normal distribution with parameters mu and sigma^2.

Proof

If Y has a normal distribution, then its probability density function is[eq4]The function[eq5]is strictly increasing, so we can use the formula for the density of a strictly increasing function[eq6]In particular, we have[eq7]so that[eq8]

Expected value

The expected value of a log-normal random variable X is[eq9]

Proof

It can be derived as follows:[eq10]where: in step $ rame{A}$ we have made the change of variable[eq11]and in step $ rame{B}$ we have used the fact that [eq12]is the density function of a normal random variable with mean $sigma $ and unit variance, and as a consequence, its integral is equal to 1.

Variance

The variance of a log-normal random variable X is[eq13]

Proof

Let us first derive the second moment [eq14]where: in step $ rame{A}$ we have made the change of variable[eq15]and in step $ rame{B}$ we have used the fact that [eq16]is the density function of a normal random variable with mean $2sigma $ and unit variance, and as a consequence, its integral is equal to 1. We can now use the variance formula [eq17]

Higher moments

The n-th moment of a log-normal random variable X is[eq18]

Proof

It can be derived as follows: [eq19]where: in step $ rame{A}$ we have made the change of variable[eq20]and in step $ rame{B}$ we have used the fact that [eq21]is the density function of a normal random variable with mean $nsigma $ and unit variance, and as a consequence, its integral is equal to 1.

Moment generating function

The log-normal distribution does not possess the moment generating function.

Characteristic function

A closed formula for the characteristic function of a log-normal random variable is not known.

Distribution function

The distribution function [eq22] of a log-normal random variable X can be expressed as[eq23]where [eq24] is the distribution function of a standard normal random variable.

Proof

We have proved above that a log-normal variable X can be written as[eq25]where Y has a normal distribution with mean mu and variance sigma^2. In turn, Y can be written as[eq26]where Z is a standard normal random variable. As a consequence,[eq27]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

A random variable X has a log-normal distribution with mean and variance equal to 1 and $e-1$ respectively.

What is the probability that X takes a value smaller than $e^{-1/2}$?

Solution

We have[eq28]We compute the square of the expected value[eq29]and add it to the variance:[eq30]Therefore, the parameters mu and sigma^2 satisfy the system of two equations in two unknowns[eq31]By taking the natural logarithm of both equations, we obtain[eq32]Subtracting the first equation from the second, we get[eq33]Then, we use the first equation to obtain [eq34]We then work out the formula for the distribution function of a log-normal variable:[eq35]In the last step we have used the fact that the distribution function [eq36] of a standard normal random variable is symmetric around zero.

How to cite

Please cite as:

Taboga, Marco (2021). "Log-normal distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/log-normal-distribution.

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