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Concept
Log-normal distribution
Summary
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.
Let be a standard normal variable, and let and be two real numbers. Then, the distribution of the random variable
is called the log-normal distribution with parameters and . These are the expected value (or mean) and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of itself.
This relationship is true regardless of the base of the logarithmic or exponential function: if is normally distributed, then so is for any two positive numbers . Likewise, if is log-normally distributed, then so is , where .
In order to produce a distribution with desired mean and variance , one uses
and
Alternatively, the "multiplicative" or "geometric" parameters and can be used.
Official source
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