What is the maximum entropy distribution for a continuous random variable on $[0,\infty)$ with given mean and variance?

Question

I know that for a given logmean and logstdev its the lognormal, but what about where we directly specify the mean and variance? The above seems to depend on the log-transformation to the maxent for unbounded continuous RV with given mean and variance (i.e, Normal).

Answer 1

The maximum entropy distribution is of the form $f(x) = \exp( \sum_k \lambda_k g_k(x))/Z$ where $g_k(x)$ are imposed by the restrictions (like Lagrange multipliers) and $Z$ is the normalization factor. In our case, we have two restrictions (apart from the trivial one), which give the two functions $g_1(x)=x$ (mean) and $g_2(x)=x^2$ (second moment, or variance)

Hence, the distribution is has the form $f(x) = \exp( a x -b x^2)/Z$ ($x\ge 0$ , $b>0$) which corresponds to a truncated normal.