Poisson entropy monotonicity

The information entropy $F(\lambda )$ of a Poisson distribution is strictly increasing for $\lambda >0$. Given ${\lambda }_2>{\lambda }_1$, take independent $U\sim \mathrm{Pois}({\lambda }_1)$ and $V\sim \mathrm{Pois}({\lambda }_2-{\lambda }_1)$. Their sum has Poisson distribution with mean ${\lambda }_2$, so entropy monotonicity under independent addition gives $F({\lambda }_2)\ge F({\lambda }_1)$. All these information entropies are finite, and $\mathrm{Cov}(U+V,V)=\mathrm{Var}(V)>0$ excludes equality.

// Target: probability-and-statistics.bigb