In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln X has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics). The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas. A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate X—for which the mean and variance of ln X are specified.
Definitions
Generation and parameters Let Z {\displaystyle Z} be a standard normal variable, and let μ {\displaystyle \mu } and σ {\displaystyle \sigma } be two real numbers, with σ > 0 {\displaystyle \sigma >0} . Then, the distribution of the random variable
X = e μ + σ Z {\displaystyle X=e^{\mu +\sigma Z}}
is called the log-normal distribution with parameters μ {\displaystyle \mu } and σ {\displaystyle \sigma } . These are the expected value (or mean) and standard deviation of the variable's natural logarithm, ln X {\displaystyle \ln X} , not the expectation and standard deviation of X {\displaystyle X} itself.
This relationship is true regardless of the base of the logarithmic or exponential function: If log a X {\displaystyle \log _{a}X} is normally distributed, then so is log b X {\displaystyle \log _{b}X} for any two positive numbers a , b ≠ 1 {\displaystyle a,b\neq 1} . Likewise, if e Y {\displaystyle e^{Y}} is log-normally distributed, then so is a Y {\displaystyle a^{Y}} , where 0 < a ≠ 1 {\displaystyle 0<a\neq 1} . In order to produce a distribution with desired mean μ X {\displaystyle \mu _{X}} and variance σ X 2 {\displaystyle \sigma _{X}^{2}} , one uses μ = ln μ X 2 μ X 2 + σ X 2 {\displaystyle \mu =\ln {\frac {\mu _{X}^{2}}{\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}}}}} and σ 2 = ln ( 1 + σ X 2 μ X 2 ) {\displaystyle \sigma ^{2}=\ln \left(1+{\frac {\sigma _{X}^{2}}{\mu _{X}^{2}}}\right)} . Alternatively, the "multiplicative" or "geometric" parameters μ ∗ = e μ {\displaystyle \mu ^{*}=e^{\mu }} and σ ∗ = e σ {\displaystyle \sigma ^{*}=e^{\sigma }} can be used. They have a more direct interpretation: μ ∗ {\displaystyle \mu ^{*}} is the median of the distribution, and σ ∗ {\displaystyle \sigma ^{*}} is useful for determining "scatter" intervals, see below.
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