Lognormal Distribution

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lognormal distribution

[′läg‚nȯr·məl ‚di·strə′byü·shən]
A probability distribution in which the logarithm of the parameter has a normal distribution.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Lognormal Distribution


a special type of probability distribution of random variables. If X has a normal distribution and if Y = ex, then Y will have a lognormal distribution characterized by the density

Here, m and σ are the parameters of the distribution of the variable X. The mathematical expectation Y is

mY = em + σ2/2

and the dispersion,

The sizes of particles of a crushed material (a rock, for example) and the content of many minerals in rocks obey this distribution to a good approximation.


Kolmogorov, A. N. “O logarifmicheski-normal’nom zakone raspredeleniia razmerov chastits pri droblenii.” Dokl. AN SSSR, 1941, vol. 31, issue 2, pp. 99–101.
Cramer, H. Matematicheskie metody statistiki. Moscow, 1948. (Translated from English.)
Aitchison, J., and J. A. C. Brown. The Lognormal Distribution. Cambridge, 1957.


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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The distribution of the data suggested a lognormal distribution of the LOAEL/NOAEL ratio.
Note: BMD, Benchmark Dose; BMDL, Benchmark Dose Lower Confidence Limit; BMDU, Benchmark Dose Upper Confidence Limit; BW, Body Weight; ED50, Effective Dose for a 50% response; LOAEL, Lowest Observed Adverse Effect Level; NOAEL, No Observed Adverse Effect Level; POD, Point of Departure; P50, median of the distribution = Geometric mean of lognormal distribution; P95/P50, ratio between the 95th percentile and the median of the distribution = [(Geometric standard deviation of lognormal distribution).sup.1.6449]; RfD, Reference Dose; SD, Standard Deviation; TD, Toxicodynamic; TK, Toxicokinetic; [UF.sub.L], Uncertainty Factor for LOAEL-to-NOAEL.
It was found that lognormal distribution has the highest average acceptance rates; that is, 14/16 = 87.5%.
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Mean Lognormal Lognormal value COV [lambda] [zeta] Direction factor 0.8 0.2 -0.24275 0.198042 [S.sup.2.sub.d] Seasonal factor 0.65 0.3 -0.47387 0.29356 [S.sup.2.sub.s] Factor 3.6 0.15 1.269809 0.149166 [S.sup.2.sub.b] Elasticity 200 0.2 5.278707 0.198042 modulus E Table 10: Effect of climate change to changes in reliability Present Future Increased condition prediction percentage Extreme temperature ([degrees]C) 41.9 44.7~46.9 6.7~11.9 Extreme wind speed (m/s) 38.35 39.37~40.17 2.7~4.7 Failure probability Monte Carlo simulation 0.0343 0.0484~0.0591 41.1~72.3 First-order estimation 0.00504 0.01004~0.01455 99.2~188.9 Numerical integration 0.0211 0.0325~0.0409 54.0~93.8 Other effects Correlation ([rho] = 0.1) 0.034 0.0486~0.0588 42.9~72.9 Lognormal distribution 0.016 0.0251~0.032 56.9~100
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In our interpretation the two lognormal distributions in the mix represent investors' view about two possible outcomes in the economy.
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