Log Normal Distribution
المؤلف:
Aitchison, J. and Brown, J. A. C.
المصدر:
The Lognormal Distribution, with Special Reference to Its Use in Economics. New York: Cambridge University Press, 1957.
الجزء والصفحة:
...
8-4-2021
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Log Normal Distribution
A continuous distribution in which the logarithm of a variable has a normal distribution. It is a general case of Gibrat's distribution, to which the log normal distribution reduces with 
and 
. A log normal distribution results if the variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.
The probability density and cumulative distribution functions for the log normal distribution are
where 
is the erf function.
It is implemented in the Wolfram Language as LogNormalDistribution[mu, sigma].
This distribution is normalized, since letting 
gives 
and 
, so
 |
(3)
|
The raw moments are
and the central moments are
Therefore, the mean, variance, skewness, and kurtosis excess are given by
These can be found by direct integration
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.
REFERENCES:
Aitchison, J. and Brown, J. A. C. The Lognormal Distribution, with Special Reference to Its Use in Economics. New York: Cambridge University Press, 1957.
Balakrishnan, N. and Chen, W. W. S. Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Amsterdam, Netherlands: Kluwer, 1999.
Crow, E. L. and Shimizu, K. (Ed.). Lognormal Distributions:Theory and Applications. New York: Dekker, 1988.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 123, 1951.
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