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) |

(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.

**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.

© 1996-9

1999-05-25