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 σ^{2})) for*x*> 0

- E(
*X*) = exp(μ + 1/2 σ^{2})

- Var(
*X*) = (exp(σ^{2}) - 1) · exp(2μ + σ^{2}).