A positive random variable X is lognormally distributed if ln(X) is normally distributed. The probability density function (pdf) of X is given by

f(x|µ, σ) = 1√ 2πxσ


[ − (lnx− µ)


] , x > 0, σ > 0, −∞ < µ <∞. (23.1)

Note that if Y = ln(X), and Y follows a normal distribution with mean µ and standard deviation σ, then the distribution of X is called lognormal. Since X is actually an antilogarithmic function of a normal random variable, some authors refer to this distribution as antilognormal. We denote this distribution by lognormal(µ, σ2).