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 (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

[4] 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 (sometimes called Gibrat's law).

Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.

The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method.

From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

[citation needed] The mode is the point of global maximum of the probability density function.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,[21] The partial expectation of a random variable

ProbOnto, the knowledge base and ontology of probability distributions[22][23] lists seven such forms: Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM[28] and PopED.

All remaining re-parameterisation formulas can be found in the specification document on the project website.

to all have finite variance and satisfy the other conditions of any of the many variants of the central limit theorem.

The main reason is that its variance is always finite, differently from what happen with certain Pareto distributions, for instance.

However a recent study has shown how it is possible to create a Log-Normal distribution with infinite variance using Robinson Non-Standard Analysis.

[35] For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions,

[48][49] The most efficient way to obtain interval estimates when analyzing 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.

[50]: Section 3.4 Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in an A/B test).

Comparing the medians of the two can easily be done by taking the log from each and then constructing straightforward confidence intervals and transforming it back to the exponential scale.

[b] However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop.

Plugin in the estimators to each of these parameters yields also a log normal distribution, which means that the Cox Method, discussed above, could similarly be used for this use-case:

For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that[55]

These scaling relations are 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.).

Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale.

Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "Multiplicative Central Limit Theorem".

[59] If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size.

Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal.

[citation needed] A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables.

Assuming log-normal distributions of the variables involved leads to consistent models in these cases.

[60] contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas.