For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers? 


Different sources indicated either the geometric or the Poisson distribution for this. As I am new to the topic, it would be great if someone would give me a hint or point me to a source with a thorough explanation of these issues.



Discrete Maximum Entropy Distribution with given mean
maximizing $S=\sum\_{n=0}^\infty p\_n \log p\_n$ with the constraint $\sum\_n n p\_n=\mu$ and $\sum\_n p\_n =1$ gives $p\_n = a b^n$, with Lagrange multipliers $a=1/(\mu+1)$ and $b=\mu/(\mu+1)$ determined by the constraints, so this is indeed a geometric distribution.


what is the source you are referring to that says the distribution is Poisson?


