Log-Normal Probability Density Family

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.
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Example(s):
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- 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.
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Counter-Example(s):
- an Inverse Gaussian, which is used for modeling waiting times.
- a Poisson Distribution, which is used for modeling the number of events in fixed intervals of time or space.
- an Exponential Distribution, which is used for modeling the time between events in a Poisson point process.
See: Maximum Entropy Probability Distribution, Probability Density Function, Skewness.

References

(Wikipedia, 2022) ⇒ https://en.wikipedia.org/wiki/log-normal_distribution Retrieved:2022-1-15.
- In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then $Y ln(X)$ has a normal distribution. Equivalently, if has a normal distribution, then the exponential function of , $X exp(Y)$ , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, financial returns and other metrics).
  The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.
  A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate —for which the mean and variance of $ln(X)$ are specified. ^[1]

(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.