In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, a log-normal (or lognormal) distribution is a continuous
probability distribution of a
random variable whose
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
is
normally distributed. Thus, if the random variable is log-normally distributed, then has a normal distribution.
Equivalently, if has a normal distribution, then the
exponential function of , , 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
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sciences, as well as
medicine
Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...
,
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
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 —for which the mean and variance of are specified.
Definitions
Generation and parameters
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
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
, 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. They have a more direct interpretation: is the median of the distribution, and is useful for determining "scatter" intervals, see below.
Probability density function
A positive random variable ''X'' is log-normally distributed (i.e., ), if the natural logarithm of ''X'' is normally distributed with mean and variance :
:
Let and be respectively the cumulative probability distribution function and the probability density function of the ''N''(0,1) distribution, then we have that
:
Cumulative distribution function
The cumulative distribution function is
:
where is the cumulative distribution function of the standard normal distribution (i.e., ''N''(0,1)).
This may also be expressed as follows:
:
where erfc is the complementary error function.
Multivariate log-normal
If is a multivariate normal distribution, then has a multivariate log-normal distribution. The exponential is applied elementwise to the random vector . The mean of is
:
and its covariance matrix is
:
Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.
Characteristic function and moment generating function
All moments of the log-normal distribution exist and
:
This can be derived by letting within the integral. However, the log-normal distribution is not determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero. Indeed, the expected value