Log-Normal Distribution
X ~ LogNormal(μ, σ²)
Lifetime / Skewed-Positive Family • Published July 19, 2026
The distribution for skewed, always-positive data (sizes, incomes, concentrations): history from Galton and McAlister's 1879 multiplicative-effects argument, the density and CDF obtained directly from the already-published Normal sheet via X = e^Y where Y = ln X, the classic mu-and-sigma-squared-aren't-the-mean-and-variance pitfall, why the moment generating function provably does not exist despite every ordinary moment being finite, a fully closed-form MLE inherited unchanged from the Normal sheet, and simulation via exponentiating a Normal draw.
This reference sheet requires a free account.
Sign up free to read the full Log-Normal Distribution reference sheet.
Sign up freePrefer a saved copy?
About This Reference Sheet
The distribution for skewed, always-positive data (sizes, incomes, concentrations): history from Galton and McAlister's 1879 multiplicative-effects argument, the density and CDF obtained directly from the already-published Normal sheet via X = e^Y where Y = ln X, the classic mu-and-sigma-squared-aren't-the-mean-and-variance pitfall, why the moment generating function provably does not exist despite every ordinary moment being finite, a fully closed-form MLE inherited unchanged from the Normal sheet, and simulation via exponentiating a Normal draw.
Support
x ∈ (0, ∞)
Parameters
μ ∈ ℝ (log-mean), σ > 0 (log-scale)