## Encyclopedia > Log-normal distribution

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# 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).

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