# Log-Normal Probability Density Family

(Redirected from Log-Normal Distribution)

A Log-Normal Probability Density Family is a probability density family characterized by a logarithmically transformed normal distribution.

• Context:
• It can be defined mathematically as $\displaystyle{ \mathcal{N}(\ln x;\mu,\sigma) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left[-\frac {(\ln x - \mu)^{2}} {2\sigma^{2}}\right], \ x \gt 0. }$
• It can be useful in modeling variables that are positively skewed and for which the value must be positive.
• Example(s):
• Modeling the distribution of income, which is often right-skewed.
• Modeling the distribution of particle sizes, which must be positive.
• Modeling the stock prices, which can't go below zero.
• Counter-Example(s):
• See: Maximum Entropy Probability Distribution, Probability Density Function, Skewness.

## References

### 2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Log-normal_distribution#Probability_density_function Retrieved:2016-5-9.
• A random positive variable $\displaystyle{ x }$ is log-normally distributed if the logarithm of $\displaystyle{ x }$ is normally distributed, : $\displaystyle{ \mathcal{N}(\mbox{ln}x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac {(\mbox{ln}x - \mu)^{2}} {2\sigma^{2}}\right], \ \ x\gt 0. }$ A change of variables must conserve differential probability. In particular, : $\displaystyle{ \mathcal{N}(\mbox{ln}x)d\mbox{ln}x = \mathcal{N}(\mbox{ln}x)\frac{d\mbox{ln}x}{dx}dx = \mathcal{N}(\mbox{ln}x)\frac{dx}{x} = {\ln\mathcal{N}}(x) dx, }$ where : $\displaystyle{ {\ln\mathcal{N}}(x;\mu,\sigma) = \frac{1}{ x\sigma \sqrt{2 \pi}}\exp\left[-\frac {(\mbox{ln}x - \mu)^{2}} {2\sigma^{2}}\right],\ \ x\gt 0 }$ is the log-normal probability density function.
1. Table 1, p. 221.