Log-Normal Probability Density Family

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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 [math]\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. }[/math]
    • 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.




  • (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Log-normal_distribution#Probability_density_function Retrieved:2016-5-9.
    • A random positive variable [math]\displaystyle{ x }[/math] is log-normally distributed if the logarithm of [math]\displaystyle{ x }[/math] is normally distributed, : [math]\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. }[/math] A change of variables must conserve differential probability. In particular, : [math]\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, }[/math] where : [math]\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 }[/math] is the log-normal probability density function.
  1. Table 1, p. 221.