 StatLect

# Log-normal distribution

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. ## Definition

Log-normal random variables are characterized as follows.

Definition Let be a continuous random variable. Let its support be the set of strictly positive real numbers: We say that has a log-normal distribution with parameters and if its probability density function is ## Relation to the normal distribution

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

Proposition Let be a normal random variable with mean and variance . Then the variable has a log-normal distribution with parameters and .

Proof

If has a normal distribution, then its probability density function is The function is strictly increasing, so we can use the formula for the density of a strictly increasing function In particular, we have so that ## Expected value

The expected value of a log-normal random variable is Proof

It can be derived as follows: where: in step we have made the change of variable and in step we have used the fact that is the density function of a normal random variable with mean and unit variance, and as a consequence, its integral is equal to 1.

## Variance

The variance of a log-normal random variable is Proof

Let us first derive the second moment where: in step we have made the change of variable and in step we have used the fact that is the density function of a normal random variable with mean and unit variance, and as a consequence, its integral is equal to 1. We can now use the variance formula ## Higher moments

The -th moment of a log-normal random variable is Proof

It can be derived as follows: where: in step we have made the change of variable and in step we have used the fact that is the density function of a normal random variable with mean 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 of a log-normal random variable can be expressed as where is the distribution function of a standard normal random variable.

Proof

We have proved above that a log-normal variable can be written as where has a normal distribution with mean and variance . In turn, can be written as where is a standard normal random variable. As a consequence, 