lognormal

A lognormal distribution describes a positive random variable whose natural logarithm is normally distributed. If a normal random variable Y has mean μ and standard deviation σ, and X = exp(Y), then X follows a lognormal distribution with parameters μ and σ, often denoted X ~ Lognormal(μ, σ). The support of a lognormal variable is (0, ∞).

The probability density function of X is

f(x) = (1 / (x σ sqrt(2π))) exp(- (ln x - μ)^2 / (2 σ^2)) for x > 0.

The cumulative distribution function is

F(x) = Φ((ln x - μ) / σ) for x > 0,

where Φ is the standard normal cdf.

Moments and basic statistics follow from the underlying normal. The k-th moment is E[X^k] = exp(k μ + ½ k^2 σ^2). In particular, the mean is E[X] = exp(μ + σ^2/2), and the variance is Var(X) = (exp(σ^2) - 1) exp(2μ + σ^2). The median is exp(μ), while the mode is exp(μ − σ^2). The distribution is positively skewed, with the degree of skew increasing with σ.

Lognormal models arise in contexts where values are formed by multiplicative factors, such as growth processes, incomes, or sizes of biological or industrial entities. They are used to model stock-price levels in finance, where log-returns are often assumed normal, implying prices themselves are lognormal. The sum of lognormal variables is not lognormal, which is an important consideration in aggregation. Parameterization can vary, but μ and σ commonly refer to the mean and stddev of the natural log of the variable.