= Overdispersion of nondegenerate mixed Poisson counts
{title2=$\operatorname{Var}N=\mathbb EL+\operatorname{Var}L$}
Let $L\geq0$ 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> give
$$
\mathbb EN=\mathbb EL,\qquad
\operatorname{Var}(N)=\mathbb EL+\operatorname{Var}(L).
$$
A 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> is
$$
\operatorname{Var}(L_A)=\lambda\mu^2\int\kappa(A-z)^2\,dz.
$$
This follows from <Campbell theorem>. If $A$ has positive finite <volume>, the <integral> is finite and positive: $0\leq\kappa(A-z)\leq1$ 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.
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