 Solution

Let ${\eta }_i=\alpha +\beta \log (x_i+10)+\gamma x_i$. The Poisson-lognormal random-effect model assigns each plate a random intensity ${\mu }_{ij}=e^{{\eta }_i+{\lambda }_{ij}}$. Conditional on that intensity its Poisson distribution has mean and variance both ${\mu }_{ij}$, but marginalizing over the normal distribution of the random effect givesThe law of total variance yieldsThis exceeds the mean whenever $\tau >0$. The independent plate effects supply additional heterogeneity without inducing dependence between plates conditional on the global parameters. Also $e^{{\eta }_i}$ is the median random intensity, not its marginal mean; the factor $e^{{\tau }^2/2}$ matters for population predictions.