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 $S=1$ and $M=0$. The probability density and cumulative distribution functions for the log normal distribution are

$$P(x) = {1\over Sx\sqrt{2\pi}} e^{-(\ln x-M)^2/(2S^2)}$$ (1)

$$D(x) = {1\over 2}\left[{1+\mathop{\rm erf}\nolimits \left({\ln x-M\over S\sqrt{2}}\right)}\right],$$ (2)

where $\mathop{\rm erf}\nolimits (x)$ is the Erf function. This distribution is normalized, since letting $y\equiv \ln x$ gives $dy=dx/x$ and $x=e^y$, so

$$\int_0^\infty P(x)\,dx= {1\over S\sqrt{2\pi}}\int_{-\infty}^\infty e^{-(y-M)^2/2S^2}\,dy=1.$$ (3)

The Mean, Variance, Skewness, and Kurtosis are given by

$$\mu = e^{M+S^2/2}$$ (4)

$$\sigma^2 = e^{S^2+2M}(e^{S^2}-1)$$ (5)

$$\gamma_1 = \sqrt{e^S-1}\,(2+e^{S^2})$$ (6)

$$\gamma_2 = e^{2S^2}(3+2e^{S^2}+e^{2s^2})-3.$$ (7)

These can be found by direct integration

$$\mu = {1\over S\sqrt{2\pi}} \int_0^\infty e^{-(\ln x-M)^2/2S^2}\,dx$$

$$= {1\over S\sqrt{2\pi}} \int_{-\infty}^\infty e^{-(y-M)^2/2S^2}e^y\,dy$$

$$= {1\over S\sqrt{2\pi}} \int_{-\infty}^\infty e^{-[-y+(y-M)^2/2S^2]}\,dy$$

$$= {1\over S\sqrt{2\pi}} \int_{-\infty}^\infty e^{-(-2S^2y+y^2-2yM+M^2)/2S^2}\,dy$$

$$= {1\over S\sqrt{2\pi}} \int_{-\infty}^\infty e^{-\{[y-(S^2+M)]^2+S^2(S^2+2M)\}/2S^2}\,dy$$

$$= {1\over S\sqrt{2\pi}} e^{M+S^2/2} \int_{-\infty}^\infty e^{-[y-(S^2+M)]^2/2S^2}\,dy$$

$$= e^{M+S^2/2},$$ (8)

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 Gilbrat's Distribution, Weibull Distribution

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.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 123, 1951.