The MaxEnt distribution of $x\in[0,\infty)$ with given mean is the exponential. The MaxEnt distribution of $x \in (-\infty,\infty)$ with given mean and variance is the Gaussian. Is there a MaxEnt distribution of $x \in (-\infty,\infty)$, when only the mean is given?
Update: Without loss of generality, the mean $\mu = 0$. Now on second thought I can see how this can be problematic, because any distribution symmetric around zero can be made "more entropic" by flattening it a bit more, and more, eventually approaching a uniform distribution, which is improper in $(-\infty,\infty)$.
How can I make this argument more precise and show that there is no MaxEnt distribution with given mean on $(-\infty, \infty)$? Otherwise, is there an alternative way to define a "MaxEnt" distribution with given mean on the whole real line?