## 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

`$E(X)=\mu_1^\prime = e^{\mu+\frac{1}{2}\sigma^2}$`

## Variance of Log-normal distribution

The variance of log-normal distribution is

`$\text{ Variance = } \mu_2 = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)$`

## 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

`$Q_3 = e^{\mu +0.675\sigma }$`

## Mode of Log-normal distribution

Mode of log-normal distribution is