Log-Normal Distribution

Compact study note.

Summary

One log-normal random variable is the exponential of a normal random variable. It is positive and right-skewed, with parameters inherited from the normal distribution on the log scale.^[1]

Prerequisites

• Normal Distribution
• Functions of Random Variables

Definition

If $Y\sim N(\mu ,{\sigma }^2)$ and $X=e^Y$ , then $X\sim \mathrm{LogNormal}(\mu ,{\sigma }^2)$ .

Notation and Assumptions

$\mu$ is any real number and $\sigma >0$ . These are not the mean and standard deviation of $X$ .

Parameters

$\mu \in \mathbb{R}$ and $\sigma >0$ .

Support

$(0,\mathrm{\infty })$ .

PMF or PDF

$f_X(x)=\frac{1}{x\sigma \sqrt{2\pi }}\exp [-(\ln x-\mu {)}^2/(2{\sigma }^2)]$ for $x>0$ .

CDF

$F_X(x)=\mathrm{\Phi }((\ln x-\mu )/\sigma )$ for $x>0$ .

Moments

Moments:

The MGF is not finite for any $t>0$ .

Essential Result

The distribution exists for every $\mu \in \mathbb{R}$ and $\sigma >0$ ; there is no extra condition including $m-r^2<0$ .

Small Example

If $Y\sim N(0,1)$ and $X=\exp (Y)$ , then the median of $X$ is $1$ and

Common Mistakes

• Confusing log-scale parameters with the mean and variance of $X$ .
• Claiming an ordinary MGF exists near zero.

Connections

• Normal Distribution
• Functions of Random Variables
• Probability Density Function

References