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. Likewise, 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. 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.
Probability density function
Some log-normal density functions with identical parameter but differing parameters
Cumulative distribution function
Cumulative distribution function of the log-normal distribution (with )
|MGF||defined only for numbers with a non-positive real part, see text|
|CF||representation is asymptotically divergent but sufficient for numerical purposes|
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 X for which the mean and variance of ln(X) are specified.
Given a log-normally distributed random variable and two parameters and that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of is normally distributed, and we can write as
with a standard normal variable.
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 is a positive number .
The two parameters and are not location and scale parameters for a lognormally distributed random variable X, but they are respectively location and scale parameters for the normally distributed logarithm ln(X). The quantity eμ is a scale parameter for the family of lognormal distributions.
In contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted , s.d., and in this article. The two sets of parameters can be related as (see also Arithmetic moments below)
Probability density functionEdit
A positive random variable X is log-normally distributed if the logarithm of X is normally distributed,
Let and be respectively the cumulative probability distribution function and the probability density function of the N(0,1) distribution.
Then we have
Cumulative distribution functionEdit
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.
Characteristic function and moment generating functionEdit
All moments of the log-normal distribution exist and
This 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.
The characteristic function is defined for real values of t but is not defined for any complex value of t that has a negative imaginary part, and therefore the characteristic function is not analytic at 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:
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 geometric mean of the log-normal distribution is , and the geometric standard deviation is . By analogy with the arithmetic statistics, one can define a geometric variance, , and a geometric coefficient of variation, .
Note that the geometric mean is less than the arithmetic mean. This is due to the AM–GM inequality, and corresponds to the logarithm being convex down. 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.
Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are given by
The parameters μ and σ can be obtained if the arithmetic mean and the arithmetic variance are known:
A probability distribution is not uniquely determined by the moments E[Xn] = enμ + 1/n2σ2 for n ≥ 1. 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 and medianEdit
The mode is the point of global maximum of the probability density function. In particular, it solves the equation :
The median is such a point where :
Arithmetic coefficient of variationEdit
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 with respect to a threshold is defined as
Alternatively, and using the definition of conditional expectation, it can be written as . For a log-normal random variable the partial expectation is given by:
where Φ 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 conditional expectation of a lognormal random variable X with respect to a threshold k is its partial expectation divided by the cumulative probability of being in that range:
The harmonic , geometric and arithmetic means of this distribution are related; such relation is given by
Occurrence and applicationsEdit
The log-normal distribution is important in the description of natural phenomena. This follows, because many natural growth processes are driven by the accumulation of many small percentage changes. These become additive on a log scale. If the effect of any one change is negligible, the central limit theorem says that the distribution of their sum is more nearly normal than that of the summands. When back-transformed onto the original scale, it makes the distribution of sizes approximately log-normal (though if the standard deviation is sufficiently small, the normal distribution can be an adequate approximation).
This multiplicative version of the central limit theorem is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if that's not true, the size distributions at any age of things that grow over time tends to be log-normal.
Examples include the following:
- Human behaviors
- In biology and medicine,
- 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 satisfy 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)
- In neuroscience, the distribution of firing rates across a population of neurons is often approximately lognormal. This has been first observed in the cortex and striatum  and later in hippocampus and entorhinal cortex, and elsewhere in the brain. Also, intrinsic gain distributions and synaptic weight distributions appear to be lognormal as well.
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 colloidal chemistry and polymer chemistry
- 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 social sciences and demographics
- In economics, there is evidence that the income of 97%–99% of the population is distributed log-normally. (The distribution of higher-income individuals follows a Pareto distribution.)
- In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoît Mandelbrot have argued  that log-Lévy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed, stock price distributions typically exhibit a fat tail.; the fat tailed distribution of changes during stock market crashes invalidate the assumptions of the central limit theorem.
- City sizes.
- 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.
- Particle size distributions produced by comminution with random impacts, such as in ball milling
- The file size distribution of publicly available audio and video data files (MIME types) follows a log-normal distribution over five orders of magnitude.
Extremal principle of entropy to fix the free parameter Edit
- In applications, is a parameter to be determined. In cases that there are no data to determine this parameter, it is possible to evaluate it from some universal principle. One is the entropy method. For growing processes which are governed by production and dissipation, it was shown that one can use some extremal principle of Shannon entropy to determine this parameter to be . This value can then be used to give some scaling relation between the inflexion point and maximum point of the lognormal distribution. It is shown that this relationship is determined by the base of natural logarithm, , and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are shown to be useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For instance, the lognormal function with such fits well with the size of secondary produced droplet during droplet impact  and the spreading of one epidemic disease.
Maximum likelihood estimation of parametersEdit
where is the density function 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, and , reach their maximum with the same and . 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 a normal distribution, then
- If is distributed log-normally, then is a normal random variable.
- If are independent log-normally distributed variables, and , then is also distributed log-normally:
- Let be independent log-normally distributed variables with possibly varying and parameters, and . The distribution of has no closed-form expression, but can be reasonably approximated by another log-normal distribution 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 is said to have a shifted log-normal distribution with support . , .
- 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
- This is a log-logistic distribution.
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