Log-normal distribution
Log-normal distributions are
probability distributions which are closely related to normal distributions: if
X is a normally distributed
random variable, then
exp(
X) has a log-normal distribution. In other words: the
natural logarithm of a log-normally distributed variable is normally distributed.
Random variables are log-normally distributed if they can be thought of as the product of many small independent factors, each of which multiplies the others. A typical example is the long-term return rate on a stock investment: it can be seen as the product of the daily return rates.
The log-normal distribution has probability density function
- f(x) = 1/(x σ √(2 π)) exp(-(ln x - μ)2 / (2 σ2)) for x > 0
where μ and σ are the
mean and
standard deviation of the variable's logarithm. The
expected value is
- E(X) = exp(μ + 1/2 σ2)
and the
variance is
- Var(X) = (exp(σ2) - 1) · exp(2μ + σ2).