# Lognormal Distribution as Maximum Entropy Probability Distribution

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?

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

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 $\sigma$.