Log-normal distribution


In probability theory, a log-normal 'distribution is a continuous probability distribution of a random variable whose logarithm 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 sciences, as well as medicine, economics and other topics.
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. 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, with Then, the distribution of the random variable
is called the log-normal distribution with parameters and These are the expected value 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. 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 is log-normally distributed (i.e., if the natural logarithm of is normally distributed with mean and variance
Let and be respectively the cumulative probability distribution function and the probability density function of the standard normal distribution, then we have that the probability density function of the log-normal distribution is given by:

Cumulative distribution function

The cumulative distribution function is
where is the cumulative distribution function of the standard normal distribution (i.e.,
This may also be expressed as follows:
where 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 element-wise 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 is not defined for any positive value of the argument, since the defining integral diverges.
The characteristic function is defined for real values of, but is not defined for any complex value of that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor formal series diverges:
However, a number of alternative divergent series representations have been obtained.
A closed-form formula for the characteristic function with in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by
where is the Lambert W function. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of.

Properties

Geometric or multiplicative moments

The geometric or multiplicative mean of the log-normal distribution is. It equals the median. The geometric or multiplicative standard deviation is.
By analogy with the arithmetic statistics, one can define a geometric variance,, and a geometric coefficient of variation,, has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of itself.
Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM–GM inequality and is a consequence of the logarithm being a concave function. In fact,
In finance, the term is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

Arithmetic moments

For any real or complex number, the -th moment of a log-normally distributed variable is given by
Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable are respectively given by:
The arithmetic coefficient of variation is the ratio. For a log-normal distribution it is equal to
This estimate is sometimes referred to as the "geometric CV", due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.
The parameters and can be obtained, if the arithmetic mean and the arithmetic variance are known:
A probability distribution is not uniquely determined by the moments for. That is, there exist other distributions with the same set of moments. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Mode, median, quantiles

The mode is the point of global maximum of the probability density function. In particular, by solving the equation, we get that:
Since the log-transformed variable has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of are
where is the quantile of the standard normal distribution.
Specifically, the median of a log-normal distribution is equal to its multiplicative mean,

Partial expectation

The partial expectation of a random variable with respect to a threshold is defined as
Alternatively, by using the definition of conditional expectation, it can be written as. For a log-normal random variable, the partial expectation is given by:
where is the normal cumulative distribution function. The derivation of the formula is provided in the Talk page. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.

Conditional expectation

The conditional expectation of a log-normal random variable —with respect to a threshold —is its partial expectation divided by the cumulative probability of being in that range:

Alternative parameterizations

In addition to the characterization by or, here are multiple ways how the log-normal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions lists seven such forms:
  • with mean,, and standard deviation,, both on the log-scale
  • with mean,, and variance,, both on the log-scale
  • with median,, on the natural scale and standard deviation,, on the log-scale
  • with median,, and coefficient of variation,, both on the natural scale
  • with mean,, and precision,, both on the log-scale
  • with median,, and geometric standard deviation,, both on the natural scale
  • with mean,, and standard deviation,, both on the natural scale

    Examples for re-parameterization

Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM and PopED. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.
For the transition following formulas hold and.
For the transition following formulas hold and.
All remaining re-parameterisation formulas can be found in the specification document on the project website.

Multiple, reciprocal, power

  • Multiplication by a constant: If then for
  • Reciprocal: If then
  • Power: If then for

    Multiplication and division of independent, log-normal random variables

If two independent, log-normal variables and are multiplied , the product is again log-normal, with parameters and where
More generally, if are independent, log-normally distributed variables, then

Multiplicative central limit theorem

The geometric or multiplicative mean of independent, identically distributed, positive random variables shows, for, approximately a log-normal distribution with parameters and, assuming is finite.
In fact, the random variables do not have to be identically distributed. It is enough for the distributions of to all have finite variance and satisfy the other conditions of any of the many variants of the central limit theorem.
This is commonly known as Gibrat's law.