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 an
absolutely
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,

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