= Poisson entropy monotonicity
{c}
The <information entropy> $F(\lambda)$ of a <Poisson distribution> is strictly increasing for $\lambda>0$. Given $\lambda_2>\lambda_1$, take independent $U\sim\operatorname{Pois}(\lambda_1)$ and $V\sim\operatorname{Pois}(\lambda_2-\lambda_1)$. Their sum has <Poisson distribution> with mean $\lambda_2$, so <entropy monotonicity under independent addition> gives $F(\lambda_2)\geq F(\lambda_1)$. All these <information entropies> are finite, and $\operatorname{Cov}(U+V,V)=\operatorname{Var}(V)>0$ excludes equality.
// Target: probability-and-statistics.bigb
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