## Definition of Log-normal distribution

The continuous random variable $X$ has a log-normal distribution if the random variable $Y=\ln (X)$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$. The probability density function of $X$ is \begin{aligned} f(x) & = \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2},x\geq 0 \end{aligned} In Log-normal distribution $\mu$ is called location parameter and $\sigma$ is called scale parameter.

Notation : $X\sim LN(\mu, \sigma^2)$.

## Standard Log-Normal Distribution

The standard form of log-normal distribution is obtained by taking $\mu=0$ and $\sigma =1$. The p.d.f of standard log-normal distribution is

\begin{aligned} f(x)& = \frac{1}{\sqrt{2\pi}x}e^{-\frac{1}{2}(\ln x)^2};x\geq 0 \end{aligned}

## Mean of Log-normal distribution

The mean of log-normal distribution is

## Variance of Log-normal distribution

The variance of log-normal distribution is

## Quartiles of Log-normal distribution

The first quartile $Q_1$ of log-normal distribution is

### $Q_1= e^{\mu -0.675\sigma }$

The second quartile $Q_2=\text{ median}$ of log-normal distribution is

### $Q_2 = e^{\mu -0\sigma }=e^\mu$

The third quartile $Q_3$ of log-normal distribution is

## Mode of Log-normal distribution

Mode of log-normal distribution is