The log-normal distribution is a type of probability distribution in which values are distributed around some mean value, but typically with a lower probability of extreme values. It is often used to model populations of features or characteristics when there are multiple sources of variation contributing to the natural variability of the data. The log-normal distribution imposes a limit on the probability of extreme values, making it useful in situations where outliers must be avoided.

In mathematics and statistics, the log-normal distribution is a type of continuous probability distribution where the logarithm of a random variable normally distributed. This means that when the random variable X is said to follow a normal distribution, its logarithmic value, ln(X), follows a log-normal distribution. An example of this is the distribution of wealth in a given population where individuals below a certain level of income form a majority, while those at the highest levels of wealth form a minority. The log-normal distribution offers a reasonable description of such situations.

The probability density function of the log-normal distribution is given by:

f(x)=1/(xσ√2π))e^(−[ln(x−μ)^2]/2σ^2

where μ is the mean of the log of the random variable and σ is the standard deviation of its logarithm.

The log-normal distribution is often used to model processes with a wide range of values which may occasionally display extreme values. This is because the log-normal distribution imposes a limit on the probability of extreme values compared to normal distributions which can have large variance. This is especially useful in situations where outliers must be avoided, for example when selling a large number of items and attempting to predict the total amount that will be sold.

The log-normal distribution is also useful in modelling non-negative observations such as shorelines or the lifespan of a particular species. Since the logarithm of any non-negative number is always positive, the log-normal model will always generate realistic estimates which reflect the underlying process.

In conclusion, the log-normal distribution is useful in a number of applications where extreme values need to be avoided or non-negative data needs to be modelled. This type of distribution is often used to model populations of features or characteristics when there are multiple sources of variation contributing to the natural variability of the data, or when it is necessary to accurately predict the total amount of items that will be sold. By imposing a limit on the probability of extreme values, the log-normal distribution can generate realistic estimates which more accurately reflect the underlying process.