## Log Normal Distribution

A Continuous Distribution in which the Logarithm of a variable has a Normal Distribution. It is a general case of Gilbrat's Distribution, to which the log normal distribution reduces with and . The probability density and cumulative distribution functions for the log normal distribution are

 (1) (2)

where is the Erf function. This distribution is normalized, since letting gives and , so
 (3)

The Mean, Variance, Skewness, and Kurtosis are given by
 (4) (5) (6) (7)

These can be found by direct integration
 (8)

and similarly for . Examples of variates which have approximately log normal distributions include the size of silver particles in a photographic emulsion, the survival time of bacteria in disinfectants, the weight and blood pressure of humans, and the number of words written in sentences by George Bernard Shaw.