Lognormaalina

Lognormaalina refers to the lognormal distribution, a probability model for positive-valued data in which the natural logarithm of the variable is normally distributed. If a random variable X is lognormal with parameters mu and sigma squared, then ln(X) follows a normal distribution with mean mu and variance sigma^2. Equivalently, X is lognormal if its distribution can be described by the probability density function f(x) = (1/(x sigma sqrt(2 pi))) exp(-(ln x - mu)^2/(2 sigma^2)) for x > 0. The cumulative distribution function is F(x) = Phi((ln x - mu)/sigma), where Phi is the standard normal CDF.

Key properties follow from the normal underlying: the mean of X is E[X] = exp(mu + sigma^2/2), the median is exp(mu), and the mode is exp(mu - sigma^2). The variance is Var(X) = (exp(sigma^2) - 1) exp(2 mu + sigma^2). A useful interpretation is that X = exp(Y) where Y is normal; hence the distribution of X is right-skewed and defined only on the positive real line.

Estimation and usage: Given data x_i > 0, mu and sigma^2 can be estimated from the log-transformed samples, using the sample mean and sample variance of ln(x_i), or via maximum likelihood. Lognormal models are common when data arise from multiplicative processes or positive measurements with long right tails. Applications appear in finance (stock prices), biology and environmental science (sizes, concentrations), and reliability studies.

Assessing lognormality typically involves analyzing the distribution of log-transformed data, Q-Q plots against a normal distribution, and normality tests on ln(X).