Under some conditions, a distribution is determined by all of its moments. Furthermore, there is a certain value of entropy for a given distribution. So my question is: 


1.Can I say that its entropy contains all the information of its moments? 


2.Is there any mathematical link between entropy and moments of a given distribution? Is there any theorem on this link? Or could someone give me detailed demonstration? If so, I can’t be more grateful!



Mathematical links between entropy and moments of a given distribution
If you're interested in entropy of a discrete distribution, then entropy tells you nothing about the moments (unless the random variable is constant).


First, it literally tells you nothing about the expected value or the variance since all the functions $aX+b$ have the same entropy (provided $a \neq 0$). And what's worse, we could take *any* bijection, and $f(X)$ would have the same entropy. But the moments of $f(X)$ would be all over the place depending on $f$. So I don't see too much possibility here.


And this could be made much worse. If the random variable isn't constant, then we could find lots and lots of pdfs with the same entropy. (Not hard to think of these.)


If you want to think about continuous random variables, then there are a few notions of entropy. If this is you, what did you have in mind?


