Log Normal Distribution

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 $e^{\mu -\sigma^2}$.

Conclusion

In this tutorial, you learned about theory of log-normal distribution like the probability density function, mean, variance, moments, quartiles and other properties of log-normal distribution.

To read more about the step by step examples and calculator for log-normal distribution refer the link log-normal Distribution Calculator with Examples. This tutorial will help you to understand how to calculate mean, variance of log-normal distribution and you will learn how to calculate probabilities and cumulative probabilities for log-normal distribution with the help of step by step examples.

To learn more about other probability distributions, please refer to the following tutorial:

Probability distributions

Let me know in the comments if you have any questions on log-normal Distribution and your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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