I am working on a problem where I know that the variable of concern $x$ is positive, and has no upper bound on its value and whose probability would vanish as we approach 0, $\lim_{x \rightarrow 0^+} p(x) = 0$ (I imagine a distribution that somewhat looks like a Gamma distribution perhaps, just the shape visually, not imposing any constraint mathematically).
From this question (Maximum entropy distribution $> 0$ with vanishing probability at zero?) I guess it would perhaps not be possible to find the solution with just these constraints.
To solve the problem, suppose I also add the constraint of a given fixed mean, $E(x) = m$.
The problem would then be to find the maximum entropy distribution subject to the constraints:
$$x>0$$
$$E(x) = m$$
$$\lim_{x \rightarrow 0^+} p(x) = 0$$
The solution would not be the exponential distribution because of the the additional constraint, $\lim_{x \rightarrow 0^+} p(x) = 0$. What would be the solution?
And if that is not possible, any suggestion for any not-very-restrictive constraints that can be added to ensure the existence of a solution?
