Log-normal distributionLog-normal distributions
are probability distributions
which are closely related to normal distributions: if X
is a normally distributed random variable
, then exp
) 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
- E(X) = exp(μ + 1/2 σ2)
and the variance
- Var(X) = (exp(σ2) - 1) · exp(2μ + σ2).