= Shared random rate gives dependent point counts
{title2=$\operatorname{Cov}(\Pi(A),\Pi(B))=\operatorname{Var}(R)|A||B|$}
Draw one <random variable> $R$, taking values zero and two with equal <probabilities>. Conditional on $R$, generate a <Poisson point process> on the real line with constant rate $R$. This is a <Cox process> directed by the <random measure> $R\,dx$: the same rate is shared by every window. It is a locally finite <simple point process> and a <stationary point process>. The <law of total expectation> gives $\mu(A)=\mathbb E[R]|A|=|A|$; the <law of total variance> and <law of total covariance> give, for bounded <Borel sets> $A,B$ with $A\cap B=\varnothing$,
$$
\operatorname{Var}(\Pi(A))=|A|+|A|^2,\qquad
\operatorname{Cov}(\Pi(A),\Pi(B))=|A||B|.
$$
The conditional <covariance> vanishes by <independence>, but the shared conditional means fluctuate together. This distinguishes conditional from unconditional <independence>.<Lectures on the Poisson Process by Günter Last and others>[Section 13.2, equation (13.7) and Proposition 13.6, printed p. 130: Cox mean measures and the extra directing-measure variance]{ref}
For a window of length $L$, the exact <probability mass function> is
$$
\mathbb P(N_W=k)=\frac12\mathbf1_{\{k=0\}}+\frac12 e^{-2L}\frac{(2L)^k}{k!},\qquad k=0,1,2,\ldots.
$$
In particular, its <void probability functional> is $(1+e^{-2L})/2$, compared with $e^{-L}$ for the unit-rate <Poisson point process>. Whole realizations can be empty or comparatively crowded, while their mean density remains one. This example demonstrates shared-rate <statistical dependence>, rather than a model of short-range offspring clusters.
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