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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.

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]

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]

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