# Discrete Maximum Entropy Distribution with given mean

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.

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.
• @Math.love.. --- These are the equations to solve for $a,b$ with $p_n=ab^n$: $\sum_n p_n = a/(1-b)=1$, $\sum_n np_n=ab/(1-b)^2=\mu$; the solution is $a=1/(\mu+1)$ and $b=\mu/(\mu+1)$. Feb 13 '18 at 20:32