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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 )
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.
Relation between normal and log-normal distribution. If is normally distributed, then is log-normally distributed.
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
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 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
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 t, but is not defined for any complex value of t 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:
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 (see also Coefficient of variation).
Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM-GM inequality, and corresponds to the logarithm being convex down. In fact,
This estimate is sometimes referred to as the "geometric CV" (GCV), 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 E[Xn] = en? + n2?2 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.
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:
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:
Overview of parameterizations of the log-normal distributions.
LogNormal7(?N,?N) with mean, ?N, and standard deviation, ?N, 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
For the transition following formulas hold
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
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 [divided], the product [ratio] is again log-normal, with parameters  and , where . This is easily generalized to the product of such variables.
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 , as the usual Central Limit Theorem, applied to the log-transformed variables, proves. That distribution approaches a Gaussian distribution, since decreases to 0.
If is distributed log-normally, then is a normal random variable.
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 log-normal distribution:
In the case that all have the same variance parameter , these formulas simplify to
For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.
If then is said to have a Three-parameter log-normal distribution with support ., .
where is the density function of the normal distribution . Therefore, the log-likelihood function is
Since the first term is constant with regard to ? and ?, both logarithmic likelihood functions, and , reach their maximum with the same and . Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations ,
For finite n, these estimators are biased. Whereas the bias for is negligible, a less biased estimator for is obtained as for the normal distribution by replacing the denominator n by n-1 in the equation for .
When the individual values are not available, but the sample's mean and standard deviations is, then the corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation and variance for and :
The most efficient way to analyze log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.
A basic example is given by scatter intervals: For the normal distribution, the interval contains approximately two thirds (68 %) of the probability (or of a large sample), and contain 95 %. Therefore, for a log-normal distribution,
contains 2/3, and
contains 95 %
of the probability. Using estimated parameters, the approximately the same percentages of the data should be contained in these intervals.
Confidence interval for
Using the principle, note that a confidence interval for is , where is the standard error and q is the 97.5 % quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for ,
Extremal principle of entropy to fix the free parameter
In applications, is a parameter to be determined. For growing processes balanced by production and dissipation, the use of a extremal principle of Shannon entropy shows that
This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal 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 log-normal function with such fits well with the size of secondary produced droplet during droplet impact  and the spreading of one epidemic disease.
The value is used to provide a probabilistic solution for the Drake equation.
Occurrence and applications
The log-normal distribution is important in the description of natural phenomena. In a prototype case, a justification runs as follows: 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.
A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.
Even if none of these justifications apply, the log-normal distribution is often a plausible and empirically adequate model. Examples include the following:
The length of comments posted in Internet discussion forums follows a log-normal distribution.
Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.
The length of chess games tends to follow a log-normal distribution.
Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.
Rubik's Cube solves, both general or by person, appear to be following a log-normal distribution.
Measures of size of living tissue (length, skin area, weight).
For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal 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.
The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
The PacBio sequencing read length follows a log-normal distribution.
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 log-normal. 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 log-normal as well.
In reliability analysis, the log-normal 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 log-normal distribution.
Particle size distributions produced by comminution with random impacts, such as in ball milling.
In computer networks and Internet traffic analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.
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