The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.
A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many independentrandom variables each of which is positive. (This is justified by considering the central limit theorem in the log-domain.) For example, in finance, the variable could represent the compound return from a sequence of many trades (each expressed as its return + 1); or a long-term discount factor can be derived from the product of short-term discount factors. In wireless communication, the delay caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed: see log-distance path loss model.
Given a log-normally distributed random variable X and two parameters μ and σ that are, respectively, the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed), we can write X as
This relationship is true regardless of the base of the logarithmic or exponential function. If loga(Y) is normally distributed, then so is logb(Y), for any two positive numbers a, b ≠ 1. Likewise, if is log-normally distributed, then so is , where is a positive number ≠ 1.
On a logarithmic scale, μ and σ can be called the location parameter and the scale parameter, respectively.
In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted m, s.d., and v in this article. The two sets of parameters can be related as (see also Arithmetic moments below)
Characteristic function and moment generating function
All moments of the log-normal distribution exist and it holds that: (which can be derived by letting within the integral). However, the expected value is not defined for any positive value of the argument as the defining integral diverges. In consequence the moment generating function is not defined. The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.
Similarly, the characteristic function E[eitX] is not defined in the half complex plane and therefore it is not analytic in the origin. In consequence, 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 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 .
The arithmetic mean, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable are given by
The location () and scale () parameters can be obtained if the arithmetic mean and the arithmetic variance are known; it is simpler if is computed first:
For any real or complex number , the thmoment of a log-normally distributed variable is given by
A log-normal distribution is not uniquely determined by its moments E[Xk] for k ≥ 1, that is, there exists some other distribution with the same moments for all k. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.
The arithmetic coefficient of variation is the ratio (on the natural scale). For a log-normal distribution it is equal to
Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.
The partial expectation of a random variable X with respect to a threshold k is defined as where is the probability density function of X. Alternatively, and using the definition of conditional expectation, it can be written as g(k)=E[X | X > k]*P(X > k). For a log-normal random variable the partial expectation is given by:
Where Phi is the normal cumulative distribution function. The derivation of the formula is provided in the discussion of this Wikipedia entry. 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.
The log-normal distribution is important in the description of natural phenomena. The reason is that for many natural processes of growth, growth rate is independent of size. This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. It can be shown that a growth process following Gibrat's law will result in entity sizes with a log-normal distribution. Examples include:
Measures of size of living tissue (length, skin area, weight);
For highly communicable epidemics, such as SARS in 2003, if publication intervention is involved, the number of hospitalized cases is shown to satistfy the lognormal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production.
The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;
Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations)
Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting
Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.
In hydrology, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.
The image on the right illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.
In reliability analysis, the lognormal distribution is often used to model times to repair a maintainable system.
In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."  Also, the random obstruction of radio signals due to large buildings and hills, called shadowing, is often modeled as a lognormal distribution.
It has been proposed that coefficients of friction and wear may be treated as having a lognormal distribution 
In spray process, such as droplet impact, the size of secondary produced droplet has a lognormal distribution, with the standard deviation ： determined by the principle of maximum rate of entropy production If the lognormal distribution is inserted into the Shannon entropy expression and if the rate of entropy production is maximized (principle of maximum rate of entropy production), then σ is given by : and with this parameter the droplet size distribution for spray process is well predicted. It is an open question whether this value of σ has some generality for other cases, though for spreading of communicable epidemics, σ is shown also to take this value.
Particle size distributions produced by comminution with random impacts, such as in ball milling
where by ƒL we denote the probability density function of the log-normal distribution and by ƒN that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ℓL and ℓN, reach their maximum with the same μ and σ. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that
If is distributed log-normally, then is a normal random variable.
If are nindependent log-normally distributed variables, and , then Y is also distributed log-normally:
Let be independent log-normally distributed variables with possibly varying σ and μ parameters, and . The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal distribution Z at the right tail. Its probability density function at the neighborhood of 0 has been characterized and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematical justified by Marlow) is obtained by matching the mean and variance of another lognormal distribution:
In the case that all have the same variance parameter , these formulas simplify to
If , then X + c is said to have a shifted log-normal distribution with support x ∈ (c, +∞). E[X + c] = E[X] + c, Var[X + c] = Var[X].
If , then
If , then
If then for
Lognormal distribution is a special case of semi-bounded Johnson distribution
If with , then (Suzuki distribution)
A substitute for the log-normal whose integral can be expressed in terms of more elementary functions can be obtained based on the logistic distribution to get an approximation for the CDF
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