Lognormale

The lognormale, or lognormal distribution, describes a positive-valued random variable X whose logarithm is normally distributed. If ln X follows a normal distribution with mean mu and variance sigma^2, then X is lognormally distributed with parameters mu and sigma^2. Equivalently, X ~ Lognormal(mu, sigma^2).

The distribution is defined for x > 0 and has the probability density function f(x) = 1 / (x sigma sqrt(2π)) exp(- (ln x - mu)^2 / (2 sigma^2)). The cumulative distribution function is F(x) = Φ((ln x - mu)/sigma), where Φ is the standard normal CDF. Key moments include the mean E[X] = exp(mu + sigma^2/2) and the variance Var(X) = (exp(sigma^2) - 1) exp(2mu + sigma^2). Skewness and excess kurtosis are functions of sigma^2: skewness = (exp(sigma^2) + 2) sqrt(exp(sigma^2) - 1) and excess kurtosis = exp(4 sigma^2) + 2 exp(3 sigma^2) + 3 exp(2 sigma^2) - 6.

Properties of the lognormale include positivity and right skewness; larger sigma yields stronger skew and heavier tails. It often arises from multiplicative processes, where many independent positive factors multiply to produce the observed value. Typical applications appear in finance (stock prices under geometric Brownian motion), environmental science, reliability, and income or size distributions.

Estimation usually proceeds via maximum likelihood or method of moments, often after a log transform to leverage normal-theory techniques. Limitations include inapplicability to zero or negative values and sensitivity to outliers; the logtransform can aid normalization but may not always be appropriate.