Log-normal distribution

A probability distribution supported on positive values whose logarithm is normally distributed. Its right-skewed shape arises because multiplicative variation becomes additive on the logarithmic scale.

Normal distribution: A continuous probability distribution symmetric about its mean, with a bell-shaped density. Taking the exponential of a normally distributed variable produces this distribution.
Logarithmic mean: The expected value of the logarithm of a positive random variable. It is the location parameter of the underlying normal distribution.
Multiplicative process: A process in which successive changes multiply a quantity by random or fixed factors. Many such processes yield approximately normal logarithms and log-normal outcomes.
Gamma distribution: A family of continuous distributions on positive values, parameterized by shape and scale. It can resemble a log-normal distribution but does not arise by exponentiating a normal variable.
Logarithm: The inverse operation to exponentiation, converting multiplication into addition. Applying the natural logarithm turns a log-normal variable into a normal one.
Logarithmic standard deviation: The standard deviation of the logarithm of a positive random variable. It controls multiplicative spread and the degree of right skew.
Income distribution: The statistical distribution of incomes across a population. A log-normal model can describe some income ranges, though not every population or tail.
Weibull distribution: A positive-valued distribution commonly used to model lifetimes and failure times. Its tail and hazard behavior differ from those of the log-normal distribution.
Geometric Brownian motion: A stochastic process whose increments in logarithmic value follow Brownian motion with drift. Its values at fixed times are log-normally distributed.
Arithmetic mean: The sum of values divided by their count, or the corresponding expectation for a random variable. It exceeds the median when the log-normal distribution is nondegenerate.

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