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According to the wikipedia article on the lognormal distribution, the lognormal distribution is "the maximum entropy probability distribution for a random variate $X$ for which the mean and variance of $\log(X)$ is fixed".

Is there a not too complicated account of what this means and how this is derived?

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  • $\begingroup$ You didn't correctly copy the sentence from wikipedia. Insert a $\ln(X)$ between 'is' and 'fixed'. $\endgroup$
    – Glen_b
    Commented Feb 26, 2013 at 2:47
  • $\begingroup$ I've inserted a $\ln(X)$ before "is fixed", thanks for the hint. $\endgroup$
    – Pugl
    Commented Feb 26, 2013 at 7:44
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    $\begingroup$ The Wikipedia article references a clear paper by Park & Bera in which the result is derived. $\endgroup$
    – whuber ♦
    Commented Feb 26, 2013 at 15:10

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Note that the $\log(X)\sim \mbox{Normal}(\mu,\sigma^2)$, therefore they are actually making a claim regarding the normal distribution. For a normal distribution we have that

The normal distribution Normal$(\mu,\sigma^2)$ has maximum entropy among all real-valued distributions with specified mean $\mu$ and standard deviation $\sigma$.

Check this wikipedia entry for more details (including the proof of this result): http://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution#Given_mean_and_standard_deviation:_the_normal_distribution

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    $\begingroup$ Thanks a lot for this hint, it was quite helpful. Also I had seen the paper linked on wikipedia, it did not look trivial though. I have one question regarding this property: We do have biological data where the data is a set of measured lenghts, and these lengths have a lognormal distribution. Knowing that in biology cost-minimization is crucial, I wondered if the Maximum Entropy property could be linked to cost-reduction? Sorry if the question is vague! P. $\endgroup$
    – Pugl
    Commented Feb 28, 2013 at 20:00
  • $\begingroup$ This is a new question, you should then ask it as a new question, not just in comments! $\endgroup$
    – kjetil b halvorsen ♦
    Commented Jun 5, 2017 at 23:38

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