Log-normal is also written log normal or lognormal. 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 sas caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed: see log-distance path loss model.
In a log-normal distribution X, the parameters denoted μ and σ are, respectively, the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed), which means
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 and standard deviation of the non-logarithmized sample values are denoted m and s.d. in this article.
A log-normal distribution with mean m and variance v has parameters
This series representation is divergent for Re(σ2) > 0. However, it is sufficient for evaluating the characteristic function numerically at positive as long as the upper limit in the sum above is kept bounded, n ≤ N, where
and σ2 < 0.1. To bring the numerical values of parameters μ, σ into the domain where strong inequality holds true one could use the fact that if X is log-normally distributed then Xm is also log-normally distributed with parameters μm, σm. Since , the inequality could be satisfied for sufficiently small m. The sum of series first converges to the value of φ(t) with arbitrary high accuracy if m is small enough, and left part of the strong inequality is satisfied. If considerably larger number of terms are taken into account the sum eventually diverges when the right part of the strong inequality is no longer valid.
Another useful representation is available by means of double Taylor expansion of e(ln x − μ)2/(2σ2).
The geometric mean of the log-normal distribution is . Because the log of a log-normal variable is symmetric and quantiles are preserved under monotonic transformations, the geometric mean of a log-normal distribution is equal to its median.
The geometric mean (mg) can alternatively be derived from the arithmetic mean (ma) in a log-normal distribution by:
Equivalently, parameters μ and σ can be obtained if the expected value and variance are known; it is simpler if σ is computed first:
For any real or complex number s, the sthmoment of log-normal X 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 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:
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 firms (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 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." 
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.
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 without mathematical justification) is obtained by matching the mean and variance:
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
^ abcdefJohnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1994), "14: Lognormal Distributions", Continuous univariate distributions. Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, ISBN978-0-471-58495-7, MR1299979
^Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis. Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN90-70754-33-9.
^Steele, C. (2008). "Use of the lognormal distribution for the coefficients of friction and wear". Reliability Engineering & System Safety93 (10): 1574–2013. doi:10.1016/j.ress.2007.09.005.edit
^ abWu, Z.N. (2003), [ "Prediction of the size distribution of secondary ejected droplets by crown splashing of droplets impinging on a solid wall"]. Probabilistic Engineering Mechanics, Volume 18, Issue 3, July 2003, Pages 241–249. doi:10.1016/S0266-8920(03)00028-6