log-normal distribution

The p.d.f. of log-normal distribution is \[ f_X(x)=\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{lnx-\mu}{\sigma})^2} \]

The log-normal distribution is a right skewed continuous probability distribution, meaning it has a long tail towards the right. It is used for modelling various natural phenomena such as income distributions, the length of chess games or the time to repair a maintainable system and more.

If the random variable \(X\) is log-normally distributed, then \(Y=\ln(X)\) has a normal distribution. We can apply the same rule from Density of transformed random variable to prove this statement. \[ \begin{gather*} X\sim LogNormal(\mu,\sigma^2)\\ Y=lnX\\ X=\psi(Y)=e^Y\\ \Rightarrow g_Y(y)=f(\psi(y))|\frac{dx}{dy}|\\ =\frac{1}{e^y\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{y-\mu}{\sigma})^2}e^y\\ =\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{y-\mu}{\sigma})^2}\\ \Rightarrow Y\sim N(\mu,\sigma^2) \end{gather*} \]

Reference

• Log-normal Distribution - A simple explanation
• Density of transformed random variable