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Log Normal Distribution

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LogNormalDistribution

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 S=1 and M=0. 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

P(x)=1/(Ssqrt(2pi)x)e^(-(lnx-M)^2/(2S^2))
(1)
D(x)=1/2[1+erf((lnx-M)/(Ssqrt(2)))],
(2)

where erf(x) is the erf function.

It is implemented in the Wolfram Language as LogNormalDistribution[mu, sigma].

This distribution is normalized, since letting y=lnx gives dy=dx/x and x=e^y, so

int_0^inftyP(x)dx=1/(Ssqrt(2pi))int_(-infty)^inftye^(-(y-M)^2/2S^2)dy=1.
(3)

The raw moments are

mu_1^'=e^(M+S^2/2)
(4)
mu_2^'=e^(2(M+S^2))
(5)
mu_3^'=e^(3M+9S^2/2)
(6)
mu_4^'=e^(4M+8S^2),
(7)

and the central moments are

mu_2=e^(2M+S^2)(e^(S^2)-1)
(8)
mu_3=e^(3M+3S^2/2)(e^(S^2)-1)^2(e^(S^2)+2)
(9)
mu_4=e^(4M+2S^2)(e^(S^2)-1)^2(e^(4S^2)+2e^(3S^2)+3e^(2S^2)-3).
(10)

Therefore, the mean, variance, skewness, and kurtosis excess are given by

mu=e^(M+S^2/2)
(11)
sigma^2=e^(S^2+2M)(e^(S^2)-1)
(12)
gamma_1=sqrt(e^(S^2)-1)(2+e^(S^2))
(13)
gamma_2=e^(4S^2)+2e^(3S^2)+3e^(2S^2)-6.
(14)

These can be found by direct integration

mu=1/(Ssqrt(2pi))int_0^inftye^(-(lnx-M)^2/(2S^2))dx
(15)
=1/(Ssqrt(2pi))int_(-infty)^inftye^(-(y-M)^2/(2S^2))e^ydy
(16)
=e^(M+S^2/2),
(17)

and similarly for sigma^2.

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.

See also

Log-Series Distribution, Logarithmic Distribution, Weibull Distribution

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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.

Referenced on Wolfram|Alpha

Log Normal Distribution

Cite this as:

Weisstein, Eric W. "Log Normal Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogNormalDistribution.html

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