 Overdispersion of nondegenerate mixed Poisson counts

If $N\mid L$ has Poisson mean $L$, conditional expectation and the law of total variance give the display. A positive variance of the random mean prevents the unconditional count from being Poisson. For Poisson parents of density $\lambda$ and offspring mean $\mu$, the count in $A$ has excess variance $\lambda {\mu }^2\int \kappa (A-z{)}^{2}dz$, which is positive for a bounded positive-area set and a nontrivial model.