Lognormalmu

Lognormalmu refers to the mu parameter of the lognormal distribution. In this usage, mu is the location parameter of the underlying normal distribution whose exponential yields the lognormal random variable. If X follows a lognormal distribution with parameters mu and sigma^2, then the natural logarithm of X is normally distributed: ln(X) ~ N(mu, sigma^2). Equivalently, X = exp(Y) where Y ~ N(mu, sigma^2).

Key properties follow from this relationship. The mean of X is E[X] = exp(mu + sigma^2/2). The variance is Var(X) = (exp(sigma^2) - 1) exp(2 mu + sigma^2). The median of X equals exp(mu), and the mode is exp(mu - sigma^2). Because mu is the mean of ln(X), it can be interpreted as the log-scale location, and it specifically equals the natural logarithm of the median of X.

Parameter estimation and interpretation are common practical concerns. Given samples X1, X2, ..., Xn from a lognormal distribution, mu is typically estimated by mu_hat = (1/n) sum ln(Xi), while sigma^2 is estimated by the sample variance of ln(Xi). This approach reflects the normal model underlying the log transform.

Lognormalmu appears in fields where multiplicative processes are modeled, such as finance, environmental science, reliability engineering, and certain biological contexts. It is important to distinguish mu from the mean or median of X itself, which are functions of both mu and sigma. See also: lognormal distribution, normal distribution, log transformation.