Log-normal Distribution

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, since it locates the curve of the distribution, and $\sigma$ is called scale parameter, since the shape of the curve depends on the value of $\sigma$.

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

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

Moments of Log-normal distribution

The $r^{th}$ raw moment of log-normal distribution is $$\begin{aligned} \mu_r^\prime & = e^{\mu r + \frac{1}{2}r^2\sigma^2}. \end{aligned}$$

Proof

Let $Y=\log X \sim N(\mu, \sigma^2)$. So $X=e^Y\sim LN(\mu,\sigma^2)$. Hence, the $r^{th}$ raw moment of log-normal distribution is $$\begin{aligned} \mu_r^\prime & = E(X^r)\\ &=E(e^{rY})\\ & = M_Y(r)\\ &\qquad (\because\text{ the m.g.f. of $Y$ with argument $r$})\\ & = e^{\mu r + \frac{1}{2}r^2\sigma^2}. \end{aligned}$$

Hence, for $r=1$,

$$\begin{aligned} \mu_1^\prime &= e^{\mu+\frac{1}{2}\sigma^2} \end{aligned}$$

and for $r=2$, $\mu_2^\prime = e^{2\mu+2\sigma^2}$. $$\begin{aligned} \text{ Variance = } \mu_2 &= \mu_2^\prime-(\mu_1^\prime)^2\\ & = e^{2\mu+2\sigma^2}-e^{2\mu+\sigma^2}\\ & = e^{2\mu+\sigma^2}(e^{\sigma^2}-1). \end{aligned}$$

Quartiles of Log-normal distribution

The quartiles of log-normal distribution are

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

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

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

Proof

Let $X$ has log-normal distribution with parameter $\mu$ and $\sigma$. Then $Y=\log_e X \sim N(\mu, \sigma^2)$ distribution. Hence $Z=\frac{\log_e X -\mu}{\sigma}\sim N(0,1)$ distribution.

The $i^{th}$ quartile $Q_i$ is given by $$\begin{aligned} & P(X\leq Q_i) = \frac{i}{4}\\ \Rightarrow & P\bigg(\frac{\log_e X-\mu}{\sigma}\leq\frac{\log_e Q_i-\mu}{\sigma}\bigg) = \frac{i}{4}\\ \Rightarrow & P\bigg(Z\leq\frac{\log_e Q_i-\mu}{\sigma}\bigg) = \frac{i}{4}. \end{aligned}$$ For $i=1$, the first quartile $Q_1$ is given by $$\begin{aligned} & P\bigg(Z\leq\frac{\log_e Q_1-\mu}{\sigma}\bigg) = \frac{1}{4}=0.25\\ \Rightarrow & \frac{\log_e Q_1-\mu}{\sigma}= z_{0.25}\\ \Rightarrow & Q_1 = e^{\mu -0.675\sigma }. \end{aligned}$$ For $i=2$, the second quartile $Q_2=\text{ median}$ is given by $$\begin{aligned} & P\bigg(Z\leq\frac{\log_e Q_2-\mu}{\sigma}\bigg) = \frac{2}{4}=0.5\\ \Rightarrow & \frac{\log_e Q_2-\mu}{\sigma}= z_{0.5}\\ \Rightarrow & Q_2 = e^{\mu -0\sigma }=e^\mu. \end{aligned}$$ For $i=3$, the third quartile $Q_3$ is given by $$\begin{aligned} & P\bigg(Z\leq\frac{\log_e Q_3-\mu}{\sigma}\bigg) = \frac{3}{4}=0.75\\ \Rightarrow & \frac{\log_e Q_3-\mu}{\sigma}= z_{0.75}\\ \Rightarrow & Q_3 = e^{\mu +0.675\sigma }. \end{aligned}$$

Mode of Log-normal distribution

The mode of log-normal distribution is $e^{\mu - \sigma^2}$.

Proof

Mode of log-normal distribution can be obtained by solving $f^\prime(x)=0$ and $f^{\prime\prime}<0$.

The p.d.f. of log-normal distribution is $$\begin{equation*} f(x;\mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2},\; x\geq 0 \\ \end{equation*}$$ Differentiating above density function withe respect to $x$ and equating to zero, we get $$\begin{aligned} & f^\prime(x)=0\\ &\Rightarrow \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2}\times\big( -\frac{1}{2\sigma^2}2(\ln x -\mu)\big) \times \frac{1}{x}\\ & + e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2}\times\frac{1}{\sqrt{2\pi}\sigma}\big(-\frac{1}{x^2}\big) =0\\ &\Rightarrow \frac{-f(x)}{x} \bigg[\frac{\ln x - \mu +\sigma^2}{\sigma^2}\bigg]=0\\ &\Rightarrow \ln x = \mu - \sigma^2\\ &\Rightarrow x = e^{\mu - \sigma^2}. \end{aligned}$$

Also, $f^{\prime\prime}<0$. Hence the mode of the log-normal distribution is $e^{\mu -\sigma^2}$.

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