Overdispersion of nondegenerate mixed Poisson counts ($\mathrm{Var}N=\mathbb{E}L+\mathrm{Var}L$)

Let $L\ge 0$ be a finite random variable with finite second moment, and let $N\mid L$ have a Poisson distribution of mean $L$. Then conditional expectation and the law of total variance giveA nondegenerate mixing law has positive variance, so the count is overdispersed relative to a Poisson distribution with the same expected value. For example, equal mixing between means $1$ and $3$ gives count expected value $2$ and variance $3$. The finite-moment assumptions justify these ordinary variance identities; a proper mixing law can instead yield infinite count moments.

A spatial example uses Poisson point process parent locations of intensity $\lambda >0$, independent Poisson distribution offspring counts of expected value $\mu >0$, and an offspring displacement probability distribution $\kappa$. Given the parents, the count in a bounded measurable region $A$ is Poisson with random mean $L_A=\mu {\sum }_z\kappa (A-z)$. Its excess variance isThis follows from Campbell theorem. If $A$ has positive finite volume, the integral is finite and positive: $0\le \kappa (A-z)\le 1$ and its integral is the volume of $A$. Thus shared random parent locations produce extra count variation even though the offspring counts are conditionally Poisson.