The Lognormal Distribution

Posted by Beetle B. on Tue 06 June 2017

If \(Y=\ln X\) is a normal distribution, then \(X\) is log-normal.

\begin{equation*} f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma} x}\exp\left(-\frac{[\ln(x)-\mu]^{2}}{2\sigma^{2}}\right) \end{equation*}

for \(x\ge0\)

Note that \(\mu,\sigma\) do not represent mean or standard deviation of \(X\), but of \(\ln X\).

Mean: \(\exp(\mu+\frac{1}{2}\sigma{2})\)

Variance: \(\exp(2\mu+\sigma^{2})\left(e^{\sigma^{2}}-1\right)\)

\begin{equation*} F(x;\mu,\sigma)=P(X\le x)=P[\ln(X)\le\ln(x)]=\Phi\left(\frac{\ln(x)-\mu}{\sigma}\right) \end{equation*}



It is the result of the product of many independent random variables, each of which is positive (follows from the Central Limit Theorem). Useful for the accumulation of many small percentage changes.