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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 58, Iss. 9 — Sep. 1, 1968
  • pp: 1267–1272

Permanence of the Log-Normal Distribution

R. L. MITCHELL  »View Author Affiliations


JOSA, Vol. 58, Issue 9, pp. 1267-1272 (1968)
http://dx.doi.org/10.1364/JOSA.58.001267


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Abstract

The distribution of the sum of log-normal variates is shown for most cases of interest to be very accurately represented by a log-normal distribution instead of a normal or Rayleigh distribution that might be expected from the central-limit theorem. As a result, observation of the log-normal distribution for the fluctuations of flux received after propagation through a random medium can be readily explained, regardless of the size of the receiving aperture. In every case, the log-normal distribution is a better representation than the normal for the distribution of the sum of log-normal variates. However, in some cases not even the log-normal distribution is very accurate. The questions of convergence and accuracy are examined in detail.

Citation
R. L. MITCHELL, "Permanence of the Log-Normal Distribution," J. Opt. Soc. Am. 58, 1267-1272 (1968)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-58-9-1267


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References

  1. V. I. Tatarski, Wave Propagation in a Turbulent Mediun, (McGraw-Hill Book Company, New York, 1961), p. 208.
  2. See Ref. 1, p. 229.
  3. D. L. Fried, G. E. Mevers, and M. P. Keister, Jr., J. Opt. Soc. Am. 57, 787 (1967).
  4. S. K. Grace and S. N. Miller, Proc. IEEE 54, 1593 (1966).
  5. P. Beckmann, J. Res. Natl. Bur. Std. (U. S.) D68, 723 (1964).
  6. G. R. Heidbreder and R. L. Mitchell, IEEE Trans. Aerospace Electron Systems 3, 5 (1967).
  7. J. Aitchison and J. A. C. Brown, The Log-Normal Distribution (Cambridge University Press, London, 1957). 1267
  8. H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), p. 185.
  9. See Ref. 8, p. 131.
  10. See Ref. 8, p. 221.
  11. D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 10 (1967).

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