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.

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

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 variablehas a log-normal distribution with parameters and .

Proof

If has a normal distribution, then its probability density function isThe functionis strictly increasing, so we can use the formula for the density of a strictly increasing functionIn particular, we haveso that

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 variableand 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.

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 variableand 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

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 variableand 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.

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

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

The distribution function of a log-normal random variable can be expressed aswhere is the distribution function of a standard normal random variable.

Proof

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

Below you can find some exercises with explained solutions.

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

What is the probability that takes a value smaller than ?

Solution

We haveWe compute the square of the expected valueand add it to the variance:Therefore, the parameters and satisfy the system of two equations in two unknownsBy taking the natural logarithm of both equations, we obtainSubtracting the first equation from the second, we getThen, we use the first equation to obtain We then work out the formula for the distribution function of a log-normal variable:In the last step we have used the fact that the distribution function of a standard normal random variable is symmetric around zero.

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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